KINEMATICS   AND   DYNAMICS 


ELEMENTARY  EXPERIMENTAL 
MECHANICS' 


BY 


A.   WILMER    DUFF,  M.A.,  D.Sc.   (EDIN.) 

>  t 

PROFESSOR  OF  PHYSICS  IN  THE  WORCESTER  POLYTECHNIC 
INSTITUTE,  WORCESTER,  MASS. 


gorfc 
THE   MACMILLAN   COMPANY 

LONDON:  MACMILLAN  &  CO.,  LTD. 
1905 

All  rights  reserved 


COPYRIGHT,  1904,  1905, 
BY  THE  MACMILLAN  COMPANY, 

Set  up  and  electrotyped.     Published  August,  1905. 


Nortoootr 

J.  S.  Cashing  &  Co.  —  Berwick  &  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


PREFACE 

IN  this  book  an  attempt  is  made  to  combine  theory  and 
practice  as  closely  as  possible.  Success  in  teaching  is  in 
proportion  to  the  extent  to  which  the  active  initiative  of 
the  student  is  aroused,  and  nothing  is  so  effective  in  this 
respect  as  laboratory  work,  if  it  be  of  the  right  kind. 
The  use  of  the  hand  and  the  eye  affords  an  invaluable 
stimulus  to  the  imagination  and  the  reason.  Without 
such  personal  work  the  interest  awakened  by  a  good 
lecture  is  apt  to  be  superficial  and  temporary,  and  the 
preparation  insured  by  recitations  is  too  often  reluctant 
and  unfruitful.  Mechanics  is  the  most  fundamental  and 
least  attractive  part  of  physics,  and  in  the  teaching  of  it 
lectures  and  recitations  need  all  of  the  aid  that  laboratory 
work  can  supply. 

A  grasp  of  principles  is  of  more  value  to  the  average 
student  than  skill  in  measurement.  While  the  exercises 
in  this  book  have  been  chosen  chiefly  with  a  view  to  the 
elucidation  of  principles,  the  need  of  an  adequate  degree 
of  precision  in  the  necessary  measurements  has  been  kept 
in  mind.  In  most  cases  a  test  of  the  accuracy  of  the 
work  is  supplied  by  a  comparison  of  the  results  of  theory 
and  experiment.  The  course  is  not  a  substitute  for,  but  is 
preliminary  to,  a  course  in  the  more  precise  measurement 
of  physical  constants.  Its  aim  is  to  stimulate  reflection  on 
concepts  and  principles,  and  the  value  of  each  exercise  is 


Vl  PEEFACE 

in  proportion  to  the  importance  and  number  of  the  physical 
ideas  which  must  be  considered  in  performing  the  exercise. 
With  a  few  exceptions  the  exercises  have  been  tried  by 
large  classes  of  students  and  have  been  found  satisfactory. 
The  exceptions  have  been  carefully  tested  by  myself  or 
an  assistant.  The  introduction  of  numerous  original  ex- 
ercises is  due  to  a  lack  of  suitable  familiar  experiments. 
Many  well-known  exercises  have  been  omitted  either  be- 
cause they  do  not  strongly  enforce  mechanical  concepts 
and  principles  or  because  they  require  complex  or  expen- 
sive instrumental  means. 

To  serve  the  purpose  stated  above,  each  exercise  should 
follow  the  related  lecture  or  recitation  as  closely  as  pos- 
sible ;  it  will  lose  much  of  its  value  if  postponed  for 
several  weeks.  I  have  therefore  endeavored  to  choose 
exercises  that  call  for  comparatively  simple  apparatus, 
so  that  sufficient  copies  of  each  part  may  be  procured  to 
enable  all  the  students  in  a  class  (or  section)  to  work 
simultaneously  and  separately  on  each  experiment.  (The 
practice  of  having  two  or  more  students  work  together 
is  very  unsatisfactory.)  Important  parts  of  the  appa- 
ratus serve  for  a  large  number  of  exercises.  A  few  ex- 
periments which  require  apparatus  of  greater  complexity 
may,  if  necessary,  be  omitted  or  may  be  performed  by 
the  instructor  in  presence  of  the  class,  the  calculations 
being  left  to  the  latter. 

The  statements  of  theory  have  of  necessity  been  brief ; 
but  brevity  in  this  respect  is  hardly  to  be  regretted. 
Diffuseness  and  repetition  are  desirable  in  an  oral  expla- 
nation, but  a  printed  statement  can  be  reread  until  it  is 
mastered.  Diffuseness  in  a  text-book  often  defeats  its 


PREFACE  vii 

own  aim.  Bright  students  skip  prolix  explanations,  and 
others  are  often  only  puzzled  and  confused  by  what  is 
unessential;  statements  of  principles  cannot  be  predi- 
gested  by  dilution.  The  directions  for  the  experiments 
have  not  been  made  so  full  as  to  leave  nothing  to  exer- 
cise the  judgment  of  the  student.  Condensed  formulae 
for  calculation  and  tabular  forms  for  reporting  have  not 
been  supplied.  These  often  tempt  the  student  to  work 
blindly  and  confine  his  attention  to  finding  figures  to  fit 
the  formulae  and  fill  the  blanks.  The  instructor  may 
supply  such  as  he  thinks  necessary  either  in  the  lecture 
which  precedes  the  exercise  or  on  the  laboratory  black- 
board. The  topics  found  under  the  heading  "Discus- 
sion" must  be  regarded  as  mere  suggestions;  many  ques- 
tions will  be  suggested  by  the  laboratory  work  or  by  the 
subsequent  discussion  between  the  class  and  the  instructor. 
Illustrative  experiments  may  be  introduced  in  either  lec- 
ture or  discussion. 

The  apparatus  is  for  the  most  part  simple  and  readily 
constructed.  It  may  also  be  obtained  at  very  reasonable 
rates  from  the  International  Instrument  Qo.  of  Cambridge, 
Massachusetts. 

I  have  to  thank  Dr.  A.  W.  Ewell,  Assistant  Professor 
of  Physics  in  the  Worcester  Polytechnic  Institute,  for 
valuable  suggestions  and  assistance,  and  also  Mr.  C.  F. 
Howe  and  Mr.  C.  B.  Harrington,  assistants  in  physics, 
for  much  valuable  aid. 

A.  WILMER  DUFF. 

WORCESTER  POLYTECHNIC  INSTITUTE, 
WORCESTER,  MASS.,  June,  1905. 


CONTENTS 

KINEMATICS 

PAGE 
CHAPTER  1 

I.  UNITS  AND  MEASURING  INSTRUMENTS  .... 

II.  POSITION  AND  DISPLACEMENT         .....  7 

III.  VELOCITY  AND  ACCELERATION       .....  18 

IV.  PERIODIC  MOTION    ........  38 


V.    FORCE        ..........      59 


DYNAMICS 

VI.  MOMENT  OF  FORCE  ........ 

VII.  RESULTANT  OF  FORCES.    EQUILIBRIUM         .        .        .119 

VIII.  WORK  AND  ENERGY         .......    133 

IX.  PERIODIC  MOTIONS  OF  RIGID  BODIES    .        .        .        .166 

ELASTIC   SOLIDS  AND  FLUIDS 

X.    MECHANICS  OF  ELASTIC  SOLIDS     .....    185 
XI.    MECHANICS  OF  FLUIDS    .......    204: 

PROBLEMS      ........        ...    253 

TABLES  ............    259 


INDEX 


For  a  shorter  course  omit  Exercises  V-VIII,  XV,  XVI  (2), 
XVIII,  XIX,  XXII,  XXIII,  XXVII,  XXIX-XXXI,  XXXVII, 
and  §§  32,  46-49,  (proofs  of  formulae  in)  72  and  75,  76,  (part 
of)  79,  87,  107,  114,  115,  119, 120,  130,  159,  (part  of)  160, 171, 
177, 179. 


KINEMATICS 


CHAPTER  I 

UNITS   AND   MEASURING  INSTRUMENTS 

1.  Mechanics  is  the  science  of  motion  and  of  the  causes 
of  changes  in  the  motion  of  bodies.  Kinematics  is  the 
branch  of  Mechanics  which  treats  of  motion.  It  is  a 
preliminary  to  the  branch  which  treats  of  the  causes  of 
change  of  motion,  or  Dynamics. 

The  ideas  with  which  we  deal  in  Kinematics  are  those 
of  Geometry  and  Time.  Geometrical  relations  are  de- 
scribed by  means  of  lengths  of  lines  and  magnitudes  of 
angles,  and  time  relations  are  described  by  means  of  inter- 
vals of  time.  To  measure  one  of  these  we  must  compare 
it  with  a  standard  of  its  own  kind,  called 


2.  Units  of  Length.  —  The  metre  was  intended  by  those 
who  devised  it  to  be  equal  to  ^oToVoolT  °^  *^e  distance 
from  the  north  pole  to  the  equator,  along  the  meridian 
through  Paris.  While  this  derivation  of  the  metre  is  of 
historical  interest,  the  metre  is  actually  defined  as  being 
the  distance  between  two  parallel  lines  on  a  platinum- 
iridium  bar  kept  at  Sevres,  near  Paris.  Every  metre 
scale  is  intended  to  be  a  copy,  more  or  less  accurate,  of 
this  standard.  Submultiples  of  the  metre  are  the  deci- 
metre (0.1  m.),  the  centimetre  (0.01  m.),  and  the  milli- 
B  1 


2  KINEMATICS 

metre  (0.001  m.).     A  multiple,  the  kilometre  (1000  m.), 
is  used  for  stating  great  lengths.     The  centimetre  is  the 
metric  unit  mostly  used  in  Physics. 
''•The  yard  is  defined  in  the  United  States  as  Qf 6Q_  of 

«    'i  '•  i     t       <»*•  £  *•«.,.  o  » •  o  7 

the  Paris  metre.  In  Great  Britain  it  is  denned  as  the 
distance,  between.  Wo,  lines  on  a  bronze  bar  kept  at  the 
office  of  the  Exchequer  in  London. 

A  submultiple  of  the  yard,  the  foot  (J  yd.),  is  the  unit 
of  length  mostly  used  by  engineers  in  English-speaking 
countries.* 

3.  Some  Instruments  used  in  measuring  Lengths.  —  The 
beam-compass  consists  of  a  straight  rod  and  two  pointers 
movable  along  the  rod.  It  is  used  in  measuring  a  dis- 
tance when  a  scale  cannot  be  brought  into  position  to  make 


FIG.  1.  —  The  Beam-compass. 

the  measurement  directly.  The  points  are  adjusted  until 
they  coincide  with  the  ends  of  the  length  to  be  measured. 
They  are  then  clamped  in  that  position  on  the  rod  and 
the  distance  between  them  measured  by  a  scale. 

An  inexpensive  beam-compass  that  will  suffice  for  these  experi- 
ments may  be  made  from  a  brass  rod  about  30  cm.  in  length  and 
two  large-sized  electrical  "  connectors."  The  bore  of  the  connectors 
should  slightly  exceed  the  diameter  of  the  rod.  Large-sized  sew- 
ing-needles inserted  by  the  head  into  small  holes  drilled  in  the  con- 
nectors and  then  soldered  in  position  complete  the  instrument. 

*  For  the  ratios  of  metric  and  English  units,  see  Table  in  Appendix. 


UNITS  AND  MEASURING  INSTRUMENTS  6 

A  vernier  caliper  is  essentially  a  beam-compass,  the 
beam  of  which  is  graduated  and  provided  with  a  device 
called  a  vernier,  for  accurately  reading  the  fractions  of 
the  smallest  division  of  the  scale.  The  principle  of  the 
vernier  will  be  understood  from  a  study  of  Fig.  2. 
Each  unit  of  the  small  scale,  called  the  vernier,  is  (in  the 
instrument  figured)  -^  shorter  than  each  unit  of  the  main 
scale,  or  10  vernier  divisions  equal  9  scale  divisions.  Let 


.  • 

1     ! 

1       1       1 

6 
1        1       1       1 

1       1       1       1             ) 

A 

1      1 
0 

Mil 

5                          10 

FIG.  2.  — The  Vernier. 

one  end  of  the  length  to  be  measured  fall  between  two 
of  the  small  divisions  of  the  main  scale,  say  at  A,  between 
5.1  and  5.2.  Then  it  is  easily  seen  that,  if  the  third  divi- 
sion of  the  vernier  coincide  with  a  division  of  the  main 
scale,  the  distance  from  5.1  to  A  is  •£§  of  the  smallest 
division  of  the  main  scale.  Hence  the  reading  in  this 
case  is  5.13.  In  general,  if  the  vernier  be  made  so  that 
n  vernier  divisions  equal  n  —  1  scale  divisions,  then  the 

"  least  count "  of  the  vernier  will  be  -  of  a  scale  divi- 

n 

sion.  To  distinguish  it  from  another  type  of  vernier, 
the  one  just  described  is  sometimes  called  a  "  direct " 
vernier.  A  "  retrograde  "  vernier  has  a  length  equal  to 
n  + 1  scale  divisions  divided  into  n  parts  on  the  vernier; 
its  divisions  are  numbered  in  a  direction  opposite  to 
that  of  the  scale  divisions.  The  two  types  of  vernier 
are  read  in  essentially  the  same  way. 


KINEMATICS 


The  vernier  is  very  important  in  many  practical  measure- 
ments. The  simplest  way  of  mastering  its  principle  is  to 
make  an  attempt  to  construct  one.  (Exercise  1.) 

In  the  micrometer  caliper  the  device  used  for  estimating 
fractions  of  the  smallest  scale  division  depends  on  the 
fact  that  when  a  uniform  screw  travels  in  a  fixed  close- 
fitting  nut,  the  dis- 

/  ^     r|[|lj|  °M  5  I  |    tance^  the     screw 

\[j  P     Ull]  Oj  !/    advances    is    propor- 

tional to  its  rotation. 
For  example,  if  the 
"pitch"  of  the  screw 

is     0.5    mm.    it    ad- 
FiG.S.-Micrometer  Screw  Gauge.  yances    Q M    mm>    ^ 

-£$  of  a  complete  rotation.  The  linear  scale  is  attached 
to  the  nut,  the  smallest  unit  of  the  scale  being  equal  to 
the  pitch  of  the  screw.  A  circular  scale  attached  to  the 
screw  makes  it  possible  to  divide  the  smallest  unit  of 
the  linear  scale  into  as  many  parts  as  there  are  divisions 
on  the  circular  scale. 


4.  Unit  of  Time.  —  The  mean  solar  second  is  -^^ -^  of 
the  mean  solar  day,  which  is  the  average,  throughout  a 
year,   of   the   intervals    that    elapse   between    successive 
transits  of  the  sun  across  the  meridian.     The  mean  solar 
minute   equals   60  mean   solar    seconds,  and    the    mean 
solar  hour  equals  60  mean  solar  minutes. 

5.  Units  of  Angle.  —  The  radian  is  the  angle  at  the 
centre  of  a  circle  of  radius  r  subtended  by  an  arc  of 
length  r.     An  angle  at  the  centre  of  a  circle  of  radius  r 


UNITS  AND  MEASURING  INSTRUMENTS  5 

subtended  by  an  arc  of  length  a  contains  --  radians. 
Since  the  circumference  of  a  circle  is  2  TT  times  the  radius, 
4  right  angles  equal  2  TT  radians  and  1  right  angle  equals 
1  TT  radians. 

A 

The    degree    is   the   ninetieth   part   of   a   right   angle. 

Since  360°  equals  2?r  radians,  1  degree  equals  — 
radians  and  1  radian  equals  57°  .29578. 


Exercise  I.    The  Principle  of  the  Vernier 

To  construct  a  direct  vernier  to  accompany  a  scale,  a  length  equal 
to  n  —  1  of  the  smallest  units  of  the  scale  must  be  divided  into  n  parts 
on  the  vernier.  For  instance,  to  supply  an  inch  scale  divided  to 
tenths  of  an  inch  with  a  vernier  reading  to  hundredths  of  an  inch, 
take  a  strip  of  paper  about  2  in.  long  and  lay  off  on  it  very  carefully 
a  length  equal  to  T%  of  an  inch.  This  length  must  next  be  divided 
into  10  parts.  This  may  be  done  with  the  aid  of  a  piece  of  cross- 
section  paper,  provided  the  smallest  division  of  the  paper  be  less  than 
that  of  the  scale. 

A  direction  is  to  be  found  on  the  cross-section  paper  such  that  the 
distance  in  that  direction  between  two  parallel  lines  separated  by  10 
spaces  equals  T%  of  an  inch.  The  vernier  strip  being  placed  in  that 
direction,  the  two  lines  mentioned  and  the  intervening  ones  will  sub- 
divide it  into  10  parts.  Slight  dents  corresponding  to  the  points  of 
subdivision  may  be  marked  on  the  vernier  strip  by  means  of  a  sharp 
knife.  Dividing  lines  should  then  be  drawn  through  the  dents  with 
a  sharp  pencil  and  a  small  square.  The  vernier  should  then  be 
fastened  by  thumb-tacks  on  a  strip  of  wood  and  the  vernier  divisions 
numbered  in  the  proper  direction.  The  accuracy  with  which  the 
vernier  has  been  subdivided  may  be  tested  by  slipping  it  along-  the 
scale  and  noticing  whether  the  difference  between  each  scale  division 
and  each  vernier  division  seems  fairly  constant. 

A  "retrograde"  vernier  should  be  constructed  in  a  similar  way 
by  dividing  ij  of  an  inch  into  10  parts  on  the  vernier.  It  may  be 


6  KINEMATICS 

mounted  on  the  other  side  of  the  same  strip  of  wood,  and  its  divisions 
should  be  numbered  in  the  proper  direction. 

With  each  of  these  verniers  and  the  inch  scale,  two  measurements 
should  be  made  of  each  of  the  dimensions  of  several  small  regular 
blocks.  Each  measurement  will  consist  in  finding,  (1)  the  zero 
reading,  that  is,  the  point  on  the  scale  opposite  the  zero  of  the  vernier 
when  the  end  of  the  scale  and  the  end  of  the  wooden  vernier  strip 
coincide,  and  (2)  the  position  of  the  zero  of  the  vernier  when  the 
scale  is  held  vertically  on  a  smooth  plane  surface  and  the  object  is 
placed  in  position  under  the  vernier. 

A  blank  form  for  tabulating  these  measurements  should  be  devised 
and  drawn  neatly.  Every  separate  measurement  should  be  recorded. 

For  practice  in  the  use  of  the  micrometer  caliper,  some  of  the 
blocks  should  also  be  measured  by  that  instrument. 

DISCUSSION 

(a)    Comparative  merits  of  "  direct "  and  u  retrograde  "  verniers. 
(Z>)    "  Least  count "  of  a  vernier ;  how  it  depends  on  the  number  of 
divisions  of  the  vernier. 

(c)  Reading  of  barometer  vernier  and  other  verniers. 

(d)  Comparative  merits  of  the  vernier  and  the  micrometer  screw 
method  of  subdivision. 

REFERENCES 

Gray's  "Treatise  on  Physics,"  Vol.  I,  §§  8-14,  on  the  "Measurement 
of  Time." 

Stewart  and  Gee's  "Elementary  Practical  Physics,"  Part  I,  Chap- 
ters I  and  II,  on  "  Measurement  of  Length  "  and  "  Angular  Measure- 
ments." 

Encyclopaedia  Britannica,  10th  edition,  "  Weights  and  Measures." 


CHAPTER   II 

POSITION  AND  DISPLACEMENT 

6.  Position.  —  The  position  of  a  point  cannot  be  stated 
definitely  without  reference  to  some   other  point  which 
may  be  called  the  starting  point.     The  statement  of  a 
position   is  in  reality  the  description  of  a  path  leading 
from  the  starting  point  to  the  position  described.      In 
other  words,  "all  position  is  relative." 

Universal  experience  shows  that  a  complete  statement 
of  the  position  of  a  point  in  space  always  requires  the  use 
of  at  least  three  numbers.  For  instance,  to  get  from  a 
starting  point  at  sea  level  to  the  top  of  a  mountain  or  the 
bottom  of  a  mine  one  must  go  a  certain  distance  east  or 
west,  a  certain  distance  north  or  south  and  a  certain  dis- 
tance up  or  down.  Or  one  might  go  a  certain  distance  in 
a  direction  that  makes  a  certain  angle  with  the  north  and 
south  line  and  a  certain  angle  with  the  Horizontal  plane 
through  the  starting  point.  In  other  words,  space  is  of 
three  dimensions. 

7.  Rectangular  Coordinates.  —  The  three  numbers  that 
are  necessary  in  order  to  state  completely  the  position  of 
a  point  are  called  the  coordinates  of  the  point.     When 
the  numbers  are  lengths  in  some  three  directions  at  right 
angles,  the  coordinates  are  called  rectangular  coordinates, 
and   lines  in  these   three  directions  intersecting  at   the 
starting  point  are  called  the  axes  of  coordinates.     The 

7 


8  KINEMATICS 

starting  point  is  called  the  origin  of  coordinates.  Three 
rectangular  coordinates  are  usually  denoted  by  x,  y,  and 
2,  and  the  corresponding  axes  are  called  the  #-axis,  the 
#-axis,  and  the  2-axis  respectively.  One  direction  along 
an  axis  is  called  the  positive  direction  of  the  axis  and  the 
opposite  the  negative  direction,  and  values  of  #,  ?/,  and  z 
in  these  directions  are  marked  by  the  signs  +  and  — 
respectively.  Any  reader  who  is  not  quite  familiar  with 
these  ideas  should  consider  the  coordinates  of  various 
points  in  a  rectangular  room  of  length  Z,  breadth  5,  and 
height  A,  the  origin  being  taken  at  a  corner,  the  centre  of 
one  side,  and  the  centre  of  the  room  successively. 

When  all  the  points  considered  in  any  case  are  known 
to  lie  in  a  single  plane,  two  rectangular  axes  and  two 
coordinates  are  sufficient.  In  this  case  the  third  item  of 
information  is  that  which  fixes  the  position  of  the  plane. 

8.  Displacement.  —  A  displacement  is  a  change  of  posi- 
tion. A  displacement  evidently  cannot  be  clearly  specified 
without  a  statement  of  its  direction  as  well  as  of  its  mag- 
nitude. If  a  point  starts  from  A  and  arrives  at  B,  the 
magnitude  of  the  displacement  is  the  length  of  the  straight 
line  AB  and  the  direction  of  .the  displacement  is  the  direc- 
tion of  the  line  AB  drawn  from  A  to  B.  The  symbol 
AB  or  AB  may  be  used  as  an  abbreviation  of  the  phrase 
"  the  displacement  whose  length  is  AB  and  whose  direc- 
tion is  from  A  to  B." 

A  point  that  starts  from  A  and  arrives  at  B  may  have 
moved  along  the  straight  line  AB  or  it  may  have  taken 
any  irregular  path  such  as  ACDB.  Hence  a  displacement 
AB  may  be  regarded  as  the  sum  of  a  series  of  successive 


POSITION  AND  DISPLACEMENT  9 

displacements  if  the  starting  point  of  the  series  be  at  A 
and  the  ending  point  at  B  or 


This  equation  may  be  read  "  a  displacement  A  C  followed 
by  a  displacement  CD  followed  by  a  displacement  DB  is 
equivalent  to  the  displacement  AB"  Thus  the  sign  of 
addition  does  not  mean  the  addition 
of  mere  numbers  or  of  quantities 
that  may  be  represented  by  lengths 
along  the  same  line,  as  in  ordinary 
Algebra  ;  nor  does  the  sign  of 
equality  mean  equality  of  mere 
numbers  or  of  lengths.  The  addi- 
tion of  displacements  is  a  geometrical  addition. 

A  zero  displacement  is  one  which  leaves  the  position  of 
the  point  unchanged.  Since  AB  +  BA  =  AA  =  0,  it  fol- 
lows that  BA  =  —AB.  Thus  subtraction  of  a  displace- 
ment is  the  same  as  the  addition  of  an  equal  and  opposite 
displacement. 

The  displacements  we  shall  be  concerned  with  are  inde- 
pendent displacements,  the  occurrence  of  any  one  does  not 
in  any  way  interfere  with  the  occurrence  of  any  other. 
Hence  they  may  be  supposed  to  take  place  in  any  order, 
and  a  consideration  of  Fig.  4  will  make  it  clear  that  the 
result  of  the  addition  is  independent  of  the  order  of  addi- 
tion. Thus  the  addition  of  three  displacements  a,  /8,  7 
in  the  order  a,  /3,  7  is  represented  by  the  figure  ACDB, 
and  their  addition  in  the  order  a,  7,  /3  is  represented  by 
the  figure  ACEB;  the  result  in  both  cases  is  the  displace- 
ment AB.  This  would  not  be  so  unless  a  displacement 


10  KINEMATICS 

represented  by  BD  were  equally  well  represented  by  an 
equal  and  parallel  line  CE  taken  in  the  same  direction. 
All  displacements  are  considered  as  equal  ivhich  have  the 
same  magnitude  and  the  same  direction. 

9.  Geometrical  Methods  of  adding  Displacements.  — The 
following  propositions  summarize  the  preceding  and  are 
convenient  for  future  reference. 

1.  The   Triangle  Method.       The  sum  of  two  displace- 
ments AB  and  BC,  where  AB  and  BO  are  two  sides  of  a 
triangle  ABO,  is  AC. 

2.  The  Parallelogram  Method.      The  sum  of  two  dis- 
placements AB  and  A  O,  where  AB  and  A  0  are  two  sides 
of  a  parallelogram  ABDC,  is  AD. 

3.  The  Polygon  Method.     The  sum  of  any  number  of 
displacements  AB,  BO,  CD  -  NP,  where  AB,  BC,CD- 
NP  are,  sides  of  a  polygon,  is  AP. 

10.  Translation. — A  change  of  position  of  a  body  is 
called  a  translation  when  all  points  in  the  body  move  equal 
distances  in  parallel  lines,  i.e.  when  they  undergo  equal 
displacements.    When  the  displacements  are  not  equal  the 
body  is   in    rotation  or    undergoes   angular   displacement 
about  some  axis.     We  shall  postpone  the  consideration  of 
angular  displacements  for  the  present. 

11.  Addition   of    Simultaneous    Displacements.  —  A    ball 
rolled  a  certain  distance  across  the  deck  of  a  ship  while 
the  ship  moves  a  certain  distance  forward  undergoes,  rela- 
tively to  the  earth,  two  simultaneous  displacements,  one 
forward,  one  sideward.     Moreover  the  displacements  are 


POSITION  AND  DISPLACEMENT 


11 


quite  independent,  one  does  not  interfere  with  or  influence 
the  other;  the  ball  would  move  the  same  distance  side- 
ward if  the  ship  did  not  move,  and  it  would  move  the  same 
distance  forward  if  it  were  not  rolled  sideward.  The 
position  of  the  ball  at  the  end  of  a  second  is  the  same  as 
if  it  were  first  moved  forward  the  distance  the  ship  moves 
in  a  second,  and  then  moved  sideward  the  distance  it 
rolls  in  a  second.  Hence  it  is  obvious  that  simultaneous 
independent  displacements  may  be  added  as  if  they  were 
successive  displacements. 

12.    Formulae  for  Resultant  of  Two  Displacements.  —  The 

sum  of  any  number  of  displacements  is  also  called  the 
resultant  of  the  displacements  and  the  displacements  are 
said  to  be  components  of  the  resultant. 

The  magnitude  of  the  resultant  of  two  displacements 
can  readily  be  found  by  trigonometry.  Let  the  displace- 
ments be  AB  and  AC  and  their  resultant  AD.  Let  the 
magnitude  of  AB  be  dv  of  A  0  d^  and  of  AD  d. 

Then  from  the  triangle  ABD 


cos  ABD. 


If  0  be  the  angle  between  the 
positive  directions  of  AB  and 
AC,  then  ^-ABD  =  ir-0  and 
therefore 


=  d*  +  d}  +  2 


cos  6. 


FIG.  5. 


If  AB  and  A  Q  be  at  right 
angles  and  their  resultant  make  an  angle  0  with  AB,  then 


and  tan  0  = 


d 


12 


KINEMATICS 


13.  Resultant  of  Three  Rectangular  Displacements.  —  The 
resultant  of  three  rectangular  displacements  OA,  OB,  00, 
is  OD,  where  OD  is  the  diagonal  of  a  rectangular  parallele- 
piped of  which  OA,  OB,  OC,  are  intersect- 
ing edges. 

For  OB  =  AE  and  OC=ED 

and  OD=  OA  +  AE-\-ED 

=  OA+  OB+OC. 

If  the  magnitudes  of  OA,  OB,  OC,  be 
dv  d2,  dz  respectively,  and  the  magnitude  of  the  resultant 
d,  then 

14.  Resolution  of  a  Displacement  into  Components. — When 
a  displacement  is  replaced  by  components  in  given  direc- 
tions, it  is  said  to  be  "  resolved  into  components  in  those 
directions,"  or,  briefly,  "  resolved  in  those  directions."    A 
displacement  can  be   resolved   into   components   in   any 
number     of     specified     directions, 

provided  a  polygon  can  be  drawn 
of  which  one  side  represents  the 
displacement  and  the  other  sides 
are  in  the  directions  specified. 

If  a  displacement  d  be  resolved 
into  a  component  d1  in  a  direc- 
tion  making  an  angle  a  with  d,  FIQ  7 

and  a  component  d%  in  a  direction 
making  a  right  angle  with  the  first  component  d1  then 

d1  =  d  cos  a,  d%  =  d  sin  a,  and  d2  =  d^  +  d£. 


POSITION  AND  DISPLACEMENT 


13 


In  stating  these  equations,  it  is  understood  that  the 
angle  a  and  the  right  angle  that  the  direction  of  d2  makes 
with  the  direction  of  d1  are  both  measured  in  the  same 
direction  (say  counter-clockwise)  from  the  direction  of  dv 

Similarly,  a  displacement  d  may  be  resolved  into  com- 
ponents in  three  directions  at  right  angles,  namely,  dv  d2, 
dy  and,  as  is  obvious  from  Fig.  6,  c?2  =  d^  +  c?22  -+-  d^. 

15.  Analytical  Method  of  adding  Displacements.  —  Con- 
sider any  number  of  component  displacements,  dv  d^  c?3,  •••, 
in  the  same  plane,  and  let  the  angles  they  make  with  the 
positive  direction,  OX,  of  a  line  in  that  plane  be  «1,  «2,  «3,  ••• 
respectively,  the  angles 
being  all  measured  in 
the  same  direction  (e.g. 
counter-clockwise)  from 
the  positive  direction  of 
OX. 

Let  each  of  the  dis- 
placements be  resolved 
into  a  component  in  the 
direction  of  OX  and  a 
component  in  a  direc- 
tion OY,  making  a  right 
angle  with  OX.  If  the 
sum  of  the  components 
in  the  direction  of  OX  be  denoted  by  Dx,  and  the  sum  of 
the  components  along  OY  by  Dy,  then 

Dx  =  c?j  cos  «x  +  c?2  cos  «2  +  dz  cos  «3  +  •  •  •  =  2c?  cos  a. 
Dy  =  d1  sin  o£j  +  c?2  sin  «2  -f  t?3  sin  «3  H =  2d  sin  a. 


FIG.  8. 


14  KINEMATICS 

The  resultant  of  Dx  along  OX  and  Dy  along  OF  is  a 
displacement  D  making  an  angle  6  with  0^,  and  by  §  12, 

1)2  =  D*  +  D*  and  tan  6=2*- 

When  tan  0  is  positive,  the  angle  6  will  be  between  0 
and  90°  if  Dx  and  Dy  be  both  positive,  and  between  180° 
and  270°  if  Dx  and  Dy  be  both  negative.  When  tan  9  is 
negative,  0  will  be  between  90°  and  180°  if  Dx  be  negative 
and  Dy  positive,  and  between  270°  and  360°  if  Dx  be  posi- 
tive-and  Dy  negative. 

If  Dx  =  0  and  Dy=0,  then  D=0.  The  converse  is 
also  true,  for  D/  and  Dyz  cannot  be  negative :  hence,  if 
D  =  0,  then  Dx  =  Dy  =  0. 

If  the  component  displacements  be  not  in  the  same 
plane,  they  may  be  resolved  into  components  in  three 
directions,  OX,  OY,  OZ,  at  right  angles.  If  the  sums 
of  the  components  in  these  directions  be  Dx,  Dy,  Dz 
respectively,  then  the  final  resultant  D  is  such  that 
D2  =  V*  +  Dy2  +  D*.  As  before,  if  Dx,  Dy,  and  Dz  be  all 
zero,  D  will  also  be  zero,  and  conversely. 

Exercise  II.    Addition  of  Displacements 

Find  (1)  graphically  by  the  polygon  method,  (2)  by  the  analytical 
method,  the  sum  of  the  following  displacements  in  the  same  plane, 
the  angle  between  each  and  a  fixed  direction,  say  from  west  to  east, 
being  as  indicated  in  brackets : 

8  (0°),  10  (60°),  6  (115°),  4  (150°),  7  (190°),  8  (270°),  12  (350°). 

The  magnitudes  and  directions  of  the  displacements  should  be 
very  carefully  laid  off  by  a  protractor  and  scale.  The  addition 
should  be  repeated,  the  components  being  taken  in  the  reverse  order 
and  the  origin  being  the  same. 


POSITION  AND  DISPLACEMENT  15 

In  applying  the  second  method,  the  components  of  the  displace- 
ments should  be  tabulated,  all  the  ^-components  being  in  one  vertical 
column  and  all  the  ^/-components  in  another. 

DISCUSSION 

(a)  Sources  of  error  in  applying  graphical  method. 
(6)  Can  a  displacement  be  resolved  into  components  in  any  two 
assigned  directions  ?     In  any  three  ? 

(c)  Which  of  two  component  displacements  more  nearly  coincides 
in  direction  with  the  resultant  ? 

(d)  Show  how  to  resolve  a  displacement  into  two  others  of  given 
magnitudes.     When  is  this  impossible  ? 

(e)  What  is  the  resultant  of  two  displacements,  m  •  OA  and  n  •  OBI 
(/)  The  component,  in  any  direction,  of  the  resultant  of  two  dis- 
placements equals  the  sum  of  the  components  of  the  two  displace- 
ments in  that  direction. 

16.  Vector  and  Scalar  Quantities.  —  Anything  that  can 
be  measured  in  terms  of  a  unit  of  the  same  kind  is  called 
a  quantity. 

Quantities  may  be  divided  into  two  classes.  Those 
which  have  magnitude  but  not  direction  are  called  scalar 
quantities,  because  they  are  measured  merely  in  terms  of 
certain  scales  or  units.  Mass,  volume,  density,  etc.,  are 
scalar  quantities.  Quantities  which  have  direction  as 
well  as  magnitude  are  called  vector*  quantities,  because 
the  simplest  vector  quantity,  a  displacement,  may  be 
regarded  as  a  carrying  of  a  body  from  one  point  to  another. 
Other  examples  of  vector  quantities  are  velocity,  accelera- 
tion, force,  etc. 

*  Derived  from  the  root  of  the  Latin  verb  for  carry;  compare  con- 
vection. 


16  KINEMATICS 

17.  Graphical  Representation  of  Vector  Quantities.  —  A 
diagram  in  geometry  is  a  graphical  representation  of  the 
distances  and  directions  of  things  in  space.  The  lines  in 
an  accurate  diagram  are  (either  really  or  perspectively) 
proportional  in  length  to  the  distances  they  represent, 
and  the  angle  between  any  two  lines  is  equal  to  the  angle 
between  the  directions  they  represent.  An  accurate  dia- 
gram will  remain  accurate  if  all  its  dimensions  be  changed 
in  a  constant  ratio,  or  if  it  be  turned  around  into  any 
new  position. 

Precisely  similar  statements  are  true  of  any  diagram  of 
displacements  such  as  we  have  already  employed.  The 
statement  that  a  certain  line  represents  a  certain  dis- 
placement means  that  the  line  is  one  in  a  diagram  of  lines 
representing  displacements,  that  the  ratio  its  length  bears 
to  that  of  any  other  line  is  the  ratio  of  the  magnitudes 
of  the  displacements  they  represent,  and  that  the  angle  be- 
tween the  two  lines  equals  the  angle  between  the  actual  dis- 
placements. A  line  in  such  a  diagram  is  called  a  vector. 
Thus  a  vector  is  a  line  which  has  a  definite  length  and  a 
definite  direction  relatively  to  other  similar  lines  in  a  diagram. 

Any  other  vector  quantity,  e.g.  a  velocity,  is  con- 
veniently represented  in  the  same  way  by  means  of  a  line 
in  a  diagram.  Such  a  diagram  can  represent  only  vector 
quantities  of  the  same  kind,  i.e.  if  one  line  represents  a 
velocity,  all  lines  in  the  diagram  represent  velocities. 
For  brevity  we  may  indicate  any  vector  quantity,  repre- 
sented in  a  diagram,  by  the  symbol  already  used  for 
representing  a  displacement.  For  example,  "  the  velocity 
AB"  means  the  velocity  represented  in  a  diagram  of 
velocities  by  the  line  AB. 


POSITION  AND  DISPLACEMENT  17 

It  is  not  obvious  that,  because  other  vector  quantities 
may  be  represented  in  the  same  way  as  displacements, 
they  may  be  added  by  the  same  methods.  This,  however, 
is  true  of  the  vector  quantities  we  shall  be  concerned 
with,  but  before  assuming  it  we  shall  prove  that  the 
addition  may  be  reduced  to  an  addition  of  displacements. 

REFERENCES 

Clifford's  "  The  Common  Sense  of  the  Exact  Sciences,"  Chapter  IV, 
"  Position,"  §§  1-4. 

Maxwell's  "  Matter  and  Motion,"  Chapter  I. 

Clifford's  "  Elements  of  Dynamic,"  Chapter  I,  on  "  Steps." 

Love's  "  Theoretical  Mechanics,"  Chapter  II,  "  Geometry  of 
Vectors." 


CHAPTER  III 

VELOCITY   AND   ACCELERATION 

18.  Velocity.  —  The  rate  of  displacement  of  a  point  is 
called  its  velocity.  Hence,  velocity  is  a  quantity  that  has 
both  magnitude  and  direction,  that  is,  it  is  a  vector  quan- 
tity. When  the  displacements  in  equal  intervals,  how- 
ever short,  are  equal  in  both  magnitude  and  direction,  the 
velocity  is  a  constant  or  uniform  velocity.  When  this 
condition  is  not  satisfied,  the  velocity  is  variable. 

The  measure  of  a  constant  velocity  is  the  displacement 
it  produces  in  unit  time  if  it  remains  constant  that  long. 
Otherwise  it  is  the  displacement  in  the  fraction  of  a  unit, 
during  which  the  velocity  is  constant,  multiplied  by  the 
number  of  such  fractions  in  unit  time. 

Rate  of  motion  without  reference  to  the  direction  of 
the  motion  is  called  speed.  Two  ships  have  the  same 
speed  if  each  travels  10  mi.  per  hour,  but  they  have  not 
the  same  velocity  unless  they  move  in  the  same  direction. 

Velocities   and   speeds,  like    displacements,  are   essen- 
tially relative,  that  is,  we  cannot  specify  the  velocity  or 
speed  of  a  point  without  reference  to  some  other  point 
So,  too,  rest  is  only  a  relative  term. 

19.  Composition  and  Resolution  of  Constant  Velocities. — The 

resultant  of  two  constant  velocities  is  the  single  velocity 
that  would  produce  in  a  certain  time  a  displacement  equal 
to  the  sum  of  the  displacements  produced  by  the  two 

18 


VELOCITY  AND  ACCELERATION  19 

velocities.  A  bird  flying  northward  in  a  current  of  air 
that  has  an  equal  velocity  eastward  has  two  component 
velocities,  but  an  observer  at  a  distance  would  only  be 
aware  of  the  fact  that  the  bird  is  moving  northeast. 

The  measure  of  the  resultant  of  two  velocities  is  the 
resultant  displacement  in  unit  time,  that  is,  the  sum  of 
the  displacements  produced  in  unit  time  by  the  two 
velocities.  Hence  the  various  methods  (triangle,  paral- 
lelogram, polygon,  and  analytical  method)  that  may  be 
used  for  adding  displacements  may  also  be  used  for  find- 
ing the  resultant  of  velocities.  Conversely,  velocities 
may  be  resolved  into  components  as  displacements  are 
resolved. 

20.  Variable  Velocity.  —  The  mean  velocity  of  a  point  in 
any  interval  of  time  is  its  displacement  in  that  interval 
divided  by  the  interval.  This  definition  will  apply  also 
to  a  point  whose  velocity  is  constant ;  but  the  mean 
velocity  will  equal  the  constant  velocity,  since  either 
multiplied  by  the  interval  will  give  the  displacement. 

When  the  velocity  of  a  point  is  variable,  the  velocity 
of  the  point  at  any  instant,  or  its  instantaneous  velocity,  is 
the  value  approached  by  the  mean  velocity  in  an  interval 
including  that  instant,  if  the  interval  is  taken  shorter  and 
shorter  without  limit.  If  As  be  the  displacement  in  an 

As 

interval  A^,  v  =  limit  — ,  as  A£  approaches  zero.  Con- 
sider, for  example,  the  mean  velocity  of  a  train  in  10  sec., 
0.1  sec.,  0.001  sec.,  and  so  on.  The  smaller  the  interval  the 
more  the  velocity  approaches  a  definite  value,  namely,  the 
instantaneous  velocity  at  any  instant  in  the  interval. 


20  KINEMATICS 

The  instantaneous  velocity,  as  defined  above,  has  a 
definite  magnitude  and  direction  at  any  instant.  A  con- 
stant velocity  of  the  same  magnitude  and  direction  would 
be  measured  by  the  displacement  it  would  produce  in 
unit  time.  Hence  the  instantaneous  velocity  of  a  point 
equals  the  displacement  produced  in  unit  time  by  an 
equal  constant  velocity. 

To  further  illustrate  the  meaning  of  an  instantaneous 
velocity  consider  the  following  :  A  train,  A,  whose  velocity 
is  increasing,  is  passing  a  train,  B,  moving  with  a  constant 
velocity  in  the  same  direction ;  a  passenger  011  A  observes 
that  'the  velocity  of  B  seems  first  to  decrease,  then  to 
become  zero,  then  to  reverse.  At  the  instant  of  relative 
rest,  the  instantaneous  velocity  of  A  equals  the  constant 
velocity  of  B.  If  we  divide  the  displacements  of  A  and 
B  in  a  short  interval,  including  the  instant  of  relative 
rest,  by  the  length  of  the  interval,  and  suppose  the  inter- 
val indefinitely  short,  we  will  get  the  instantaneous 
velocity  of  A  and  the  equal  constant  velocity  of  B. 

21.  Composition  of  Instantaneous  Velocities.  —  An  instan- 
taneous velocity  may  be  measured  by  the  displacement  it 
would  produce  in  unit  time  if  it  continued  constant  that 
long  ;  similarly  for  a  second  instantaneous  velocity.    These 
supposed  displacements  maybe  compounded  by  the  triangle, 
parallelogram,  polygon,  or  analytical  method.     Hence  in- 
stantaneous velocities  may  be  similarly  compounded. 

22.  Acceleration.  —  A  change   of   velocity  may   be   an 
increase  of  speed,  as  in  the  case  of  a  train  leaving  a  sta- 
tion, a  decrease  of  speed,  as  in  the  case  of  a  train  approach- 
ing a  station,  or  a  change  in  the  direction  of  the  velocity, 


VELOCITY  AND  ACCELERATION  21 

as  in  the  case  of  a  train  rounding  a  curve,  or  it  may  be  a 
change  of  direction  accompanied  by  a  change  of  speed,  as 
in  the  case  of  a  train  approaching  or  leaving  a  station  on 
a  curve.  Any  such  change  of  velocity  is  called  an  incre- 
ment of  velocity.  An  increment  of  a  velocity  has  a  defi- 
nite magnitude  and  a  definite  direction,  or  it  is  a  vector 
quantity. 

The  rate  of  change  of  a  velocity  is  called  the  accelera- 
tion of  the  velocity.  When  the  increments  of  velocity  in 
all  equal  times  are  equal  as  regards  both  magnitude  and 
direction,  the  acceleration  is  called  a  constant  acceleration. 
A  constant  acceleration  may  be  measured  by  the  incre- 
ment of  velocity  to  which  it  gives  rise  in  unit  time.  Hence, 
like  increment  of  velocity  and  displacement,  acceleration 
is  a  vector  quantity. 

When  an  acceleration  is  variable  the  mean  acceleration 
and  the  instantaneous  acceleration  are  defined  as  in  the 
analogous  case  of  variable  velocity.  The  reader  should 
construct  these  definitions  for  himself. 

23.  Composition   of  Accelerations.  —  Since   accelerations 
are  measured  by  the  increments  of  velocity  to  which  they 
give  rise  in  unit  time,  accelerations  may  be  compounded 
and   resolved  as  velocities  and   displacements   are   com- 
pounded and  resolved. 

24.  Constant  Acceleration  in  the  Line  of  Motion.  —  The 

simplest  case  of  a  constant  acceleration  is  when  the  incre- 
ments of  velocity  are  in  the  direction  of  the  velocity,  that 
is,  when  they  are  increases  or  decreases  of  speed.  This 
is  the  case  of  a  body  projected  vertically  upward  or  down- 
ward or  a  train  leaving^ or  xasp^oaching  a  station  on  a 


22  KINEMATICS 

straight  horizontal  track  (though  in  the  latter  case  the 
acceleration  may  sometimes  be  variable  in  magnitude). 

If  the  constant  acceleration  be  a  and  the  velocity  at  the 
beginning  of  a  certain  interval  of  time  be  w,  then  the 
increment  of  velocity  in  the  time  t  is  at  and  the  velocity 
at  the  end  of  the  time  is 

v  =  u  4-  at.  (1) 

Since  the  velocity  increases  uniformly,  its  average  value 
in  the  interval  is  equal  to  the  velocity  at  the  middle  of 
the  interval  or  u  -f  |-  at.  Hence  the  distance  traversed  in 

time*  is  8  =  (u  +  %af)t  =  ut+%at*.  (2) 

(A  more  satisfactory  proof  of  this  equation  is  suggested 
by  (V)  in  "discussion"  of  Exercise  IV.) 
Eliminating  t  from  (1)  and  (2),  we  get 

v*  =  uz  +  2as.  (3) 

25.  Acceleration  of  Gravity.  —  It  has  been  found  by 
experimental  methods  that  the  speed  of  a  body  falling 
freely  in  a  vertical  line  increases  by  about  32.2  ft.  per 
second  in  every  second,  or  980  cm.  per  second  in  every 
second.  (These  are  only  approximate  figures,  since  the 
increase  is  slightly  different  at  different  places ;  see  Table 
in  Appendix.)  If  a  body  is  moving  vertically  upward, 
its  speed  decreases  at  the  rate  stated.  If  the  body's 
motion  is  at  some  inclination  to  the  vertical,  the  com- 
ponent of  its  velocity  vertically  downward  increases  at  the 
above  rate.  Briefly  stated,  the  acceleration  of  gravity 
is  vertically  downward,  and  its  magnitude  is  32.2  ft.  per 
second  per  second,  or  980  cm.  per  second  per  second. 


VELOCITY  AND  ACCELERATION  23 

26.  Units  and  Dimensions.  —  The  fundamental  units 
used  in  kinematics  are  the  unit  of  length  and  the  unit 
of  time.  Other  units  defined  in  terms  of  these  are  called 
derived  units.  A  system  of  units  in  which  the  derived 
units  bear  the  simplest  possible  relation  to  the  funda- 
mental units  is  called  an  absolute  system.  In  such  a 
system  the  unit  of  surface  is  the  square  of  the  unit 
of  length,  or,  as  it  may  be  briefly  expressed,  ($)  =  (jL)2. 
Similarly,  the  unit  of  volume  =  (L)3.  The  unit  of  ve- 
locity in  an  absolute  system  is  unit  length  per  unit  time, 
and  its  magnitude  therefore  varies  directly  as  the  magni- 
tude of  the  unit  of  length,  and  inversely  as  the  magnitude 
of  the  unit  of  time,  or,  briefly,  ( F)  oc  (£)  ( T)~\  The 
unit  of  acceleration  is  unit  velocity  gained  in  unit  time, 
and  its  magnitude  therefore  varies  directly  as  the  magni- 
tude of  the  unit  of  velocity,  and  inversely  as  the  magni- 
tude of  the  unit  of  time,  or  (A)  oc  ( FX^)"1 «  (^)(^)~2- 
These  relations  are  called  dimensional  relations.  They 
may  be  expressed  in  words  thus :  velocity  is  of  1  dimen- 
sion in  length  and  —  1  dimension  in  time ;  accelera- 
tion is  of  1  dimension  in  length  and  '—  2  dimensions 
in  time. 

The  numerical  measure  of  any  quantity  varies  inversely 
as  the  magnitude  of  the  unit  of  the  same  kind  in  which  it 
is  measured.  Thus,  a  length  that  is  3  in  feet  is  1  in  yards, 
and  a  velocity  that  is  3  in  feet  per  second  is  1  in  yards  per 
second.  These  are  simple  cases  of  changes  of  units ;  more 
complicated  cases  are  most  readily  worked  out  by  means 
of  dimensional  relations.  For  instance,  suppose  an  accel- 
eration of  32.2  ft.  per  second  per  second  is  to  be  expressed 
in  metres  per  minute  per  minute.  The  dimensional  rela- 


24  KINEMATICS 

tion  of  acceleration  to  length  and  time  is 
and  the  numerical  measure  of  the  acceleration  varies 
inversely  as  the  unit  of  acceleration.  Denoting  the 
measure  of  the  acceleration  in  metres  per  minute  per 
minute  by  #, 

x          (foot)  (sec.)"2      _   (foot)      (min.)2  =  30.48* 
32.2  "(metre) (min.)-2 "(metre)  '  (sec.)2  ~    100 

z  =  35.3x!04. 

(The  student  should  note  in  the  statement  of  the  result 
the  use  of  powers  of  10  to  express  large  numbers  and  also 
that,  since  32.2  is  only  given  to  one-third  of  one  per  cent, 
it  is  not  worth  while  calculating  x  to  a  higher  degree  of 
apparent  accuracy.) 

27.  Motion  of  a  Point  that  has  a  Constant  Velocity  in 
one  direction  and  a  Constant  Acceleration  in  a  direction 
at  right  angles.  —  A  consideration  of  any  such  case  will 
make  it  clear  that  the  two  parts  of  the  motion  may  be 
considered  as  taking  place  separately  and  independently. 
Thus,  a  ball  thrown  upward  in  a  train  moving  uniformly 
returns  to  the  hand.  Relatively  to  the  earth  the  ball 
describes  a  curve,  but  throughout  its  motion  it  keeps  its 
horizontal  velocity  unchanged,  and  its  vertical  motion  is 
the  same  as  if  it  had  no  horizontal  velocity.  When  the 
velocity  of  the  train  is  changing  rapidly,  the  ball  does  not 
return  to  the  hand.  The  reader  should  think  this  illus- 
tration out  carefully,  assuming  different  values  for  the 
acceleration  of  the  train  and  the  time  the  ball  is  in  the  air. 

*  1  ft.  =  30.48  cm.  approximately. 


VELOCITY  AND  ACCELERATION 


25 


When  a  body  is  projected  from  the  earth  in  any  direc 
tion,  its  horizontal  displace- 
ment, #,  at  any  time,  £,  and 
the  vertical  displacement,  #,  at 
the  same  time  may  be  calcu- 
lated separately.  If  v1  be  the 
horizontal  component  of  the 
velocity  of  projection,  and  v2 
the  component  vertically  up- 
ward, then,  since  the  accelera- 
tion of  gravity,  g,  is  vertically  downward, 

x  =  v- 


1>2 


FIG.  9. 


FIG.  10. 


From  these  two  equations  we  may  eliminate  t  and  get 
an  equation  connecting  x  and  y  that  holds  true  for  any 

simultaneous  values   of    x   and 

X    y.     This  equation  is  called  the 

equation  of  the  curve  which 
the  body  describes.  If  the 
body  be  projected  horizontally 
from  an  elevation,  then  v2  =  0 
and  vl  equals,  the  velocity  of 

projection.  If  the  positive  direction  of  y  be  taken  down- 
ward, x  =  vj  and  y  —  \  gt2.  In  this  case  the  constant 
relation  between  x  and  y  may  be  written 

—  =  — -  (which  is  a  constant). 

The   horizontal   velocity   at   any  time   is   v^   and   the 
vertical  velocity  v2  =  gt.      If   V  be   the   resultant  velo- 


26  KINEMATICS 

city  at  time  £,  and  0  the  angle  it  makes  with  the  hori- 
zonta1' 


Exercise  III.     Path  of  a  Projectile 

Apparatus.  —  A  cross-sectioned  board*  is  mounted  in  a  vertical 
plane  with  its  lines  horizontal  and  vertical  respectively.  At  one  of 
the  upper  corners  of  the  board  a  block  in  which  there  is  a  curved 
groove  is  attached  to  the  board  in  such  a  way  that  it  is  adjustable  in 
a  vertical  plane.  A  steel  ball  rolling  down  the  groove  is  projected  in  a 
horizontal  direction,  and  after  describing  a  curved  path  in  front  of  the 
board  is  caught  in  a  small  bag  or  pocket.  A  simple  spring  release 
worked  by  a  cord  enables  the  observer  to  drop  the  ball  while  he  is 
standing  in  front  of  the  board  in  a  position  convenient  for  observing 
the  falling  ball. 

Observation  of  Curve  of  Descent.  —  The  apparatus  should  first  be 
adjusted  by  means  of  a  level  so  that  one  set  of  lines  is  truly  hori- 
zontal and  the  lower  straight  end  of  the  "shoot"  is  also  horizontal. 
These  adjustments  should  be  occasionally  retested.  It  is  necessary 
that  in  successive  observations  the  release  should  be  worked  as  uni- 
formly as  possible.  For  this  purpose  a  small  weight  is  attached  to 
the  end  of  the  cord  ;  when  the  ball  is  to  be  dropped  the  small  weight 
is  pushed  off  a  platform  and,  falling  three  or  four  centimetres,  jerks 
the  cord  and  releases  the  ball.  Another  simple  method  that  some- 
times gives  more  satisfactory  results  is  to  remove  the  platform  that 
supports  the  weight  and  raise  the  weight,  by  means  of  the  cord,  to 
some  definite  position  and  allow  it  to  fall  three  or  four  centimetres. 
Care  must,  however,  be  taken  that  the  fall  is  not  so  great  as  to  alter 
the  spring  that  holds  the  ball. 

The  intersection  of  the  curve  of  descent  with  each  horizontal  line 
of  the  board  is  to  be  observed  about  six  times  in  as  many  successive 

*  The  ruled  board  referred  to  in  this  and  subsequent  exercises  may 
with  some  advantage  be  replaced  by  a  plain  board  on  which  a  sheet  of 
cross-section  paper  is  fastened  by  thumb-tacks.  The  board  will  prob- 
ably shrink  somewhat,  and  the  shrinkage  will  be  different  across  and  along 
the  grain. 


VELOCITY  AND  ACCELERATION 


27 


falls  and  the  mean  taken.     Each  reading  should  be  estimated  to  one- 
tenth  of  the  smallest  division  of  the  horizontal  line.     The  observer 


FIG.  11. 


should  endeavor  to  avoid  "  parallax  "  by  holding  his  head  in  such  a 
position  at  the  moment  of  observation  that  the  gaze  is  fixed  at  right 


28  KINEMATICS 

angles  to  the  board.  The  lines  may  be  taken  in  any  order  found  con- 
venient. There  is  some  advantage  in  taking  the  first,  third,  fifth,  etc., 
lines  and  afterward  the  second,  fourth,  etc.  In  this  way  a  complete 
curve  may  be  obtained  even  if  time  does  not  permit  all  the  lines  to  be 
observed.  Moreover,  the  observer  becomes  more  expert  with  practice 
and  the  increased  accuracy  will  be  more  evenly  distributed. 

All  these  readings  should  be  arranged  in  tabular  form,  the  readings 
referring  to  any  one  horizontal  line  being  in  a  vertical  column.  When 
the  readings  have  been  completed,  the  mean  values  should  be  plotted  on 
cross-section  paper,  the  origin  being  taken  at  an  upper  corner  of  the 
paper  in  imitation  of  the  position  of  the  board.  A  smooth  curve 
should  then  be  drawn,  striking  an  average  path  among  the  points 
located  but  not  necessarily  passing  through  any  particular  point.  For 
drawing  the  curve  celluloid  curve-forms  should  be  used. 

x2 
Calculation  of  Initial  Velocity.  —  The  constancy  of  —  may  be  tested 

by  means  of  the  values  obtained  for  each  horizontal  distance  and  the 
corresponding  vertical  distance.  Before  this  is  done  care  should  be 
taken  to  ascertain  whether  the  ball  at  the  moment  of  discharge  from 
the  "  shoot "  was  exactly  at  the  origin  of  the  cross  lines  of  the  board. 
If  not,  allowance  for  the  initial  position  of  the  ball  must  be  made  by 
subtracting  its  initial  vertical  and  horizontal  distances  from  the  mean 
values  recorded  for  the  other  points  on  the  curve.  Moreover,  if  the 
unit  of  the  board  is  not  a  centimetre,  the  corrected  horizontal  and  ver- 
tical distances  should  be  reduced  to  centimetres  before  the  calculations 

#2 
of  —  are  made. 

y 

The  initial  velocity  of  the  ball  is  calculated  from  the  final  mean 

™2 

value  of  — 

DISCUSSION 

(a)  Meaning  and  derivation  of  formulae. 

(6)  Consider  the  motion  of  a  line  passing  through  the  ball  in  the 
experiment  and  through  another  ball  discharged  simultaneously  with 
the  same  velocity  in  the  same  direction  but  supposed  devoid  of  weight. 

(c)  Consider  the  motion  of  a  line  passing  through  the  ball  and 
through  another  ball  allowed  to  fall  simultaneously  from  the  same  point. 


VELOCITY  AND  ACCELERATION  29 

(d)  At  what  point  on  the  curve  and  at  what  time  was  the  direc- 
tion of  motion  of  the  ball  inclined  at  45°  to  the  horizontal  ?    At  30°  ? 
At  60°? 

(e)  Show  that  F2  increases  as  if  the  fall  were  wholly  vertical. 
(/)  From  what  height  would  the  ball  have  to  fall  freely  to  attain 

its  initial  velocity? 

(g)  Could  the  initial  velocity  be  deduced  from  the  height  of  the 
groove  ? 

(h)  Does  the  rotation  of  the  ball  affect  the  results? 

(i)  Equations  of  the  vertical  and  horizontal  motion  and  of  the 
path  when  a  ball  is  discharged  obliquely. 

0)  What  initial  velocity  must  a  bullet  have  to  fall  back  to  its 
starting  point  in  10  sec.  ? 

(&)  A  body  is  projected  at  an  angle  of  30°  with  the  horizontal  with 
a  velocity  of  30  m.  per  second.  When  and  where  will  it  again  meet 
the  horizontal  plane  through  the  starting  point  ?  How  high  will  it 
ascend  ?  (First  find  how  long  the  vertical  motion  lasts.) 

(/)  What  is  the  final  speed  of  a  body  which,  moving  with  uniform 
acceleration,  travels  72  m.  in  2  min.  if : 

(1)  the  initial  speed  =  0? 

(2)  the  initial  speed  =  15  cm.  per  second? 

(m)  At  what  angle  with  the  shore  must  a  boat  be  directed  in  order 
to  reach  a  point  on  the  other  shore  directly  opposite  if  the  speed  with 
which  the  boat  is  rowed  be  4  mi.  an  hour  and  that  of  the  stream  3  mi. 
an  hour  ? 

(n)  A  raindrop,  falling  nearly  vertically  in  still  air,  makes  a 
"  streak  "  inclined  at  30°  to  the  vertical  on  the  pane  of  a  railway  car 
travelling  at  25  mi.  an  hour.  What  is  the  velocity  of  the  drop  ? 

28.  Curve  of  Speed.  —  A  curve  of  speed  is  a  convenient 
method  of  showing  graphically  the  way  in  which  the  speed 
of  a  body  varies.  A  horizontal  line  OT  is  drawn  to  rep- 
resent time  reckoned  from  some  moment  represented  by 
0.  At  the  end  T^  of  a  length  OT^  that  represents  an 
interval  ^  an  ordinate  TlSl  is  erected  to  represent  the 


30 


KINEMATICS 


T, 


T3 
Fia.  12. 


speed  at  the  end  of  the  interval  tr  Similar  ordinates  are 
erected  at  points  T%,  Tz  .  .  .  Tn.  A  smooth  curve  through 
jSv  $2,  $3  •••  Sn  is  the  curve  of  speed  of  the  moving  point. 
When  the  curve  has  been  drawn  the  speed  at  the  end  of 

any  interval  t  can  be 

S3-— - rr"~~"T r  found   by  measuring 

S>\^<^ "  the    ordinate   at   the 

corresponding  point 
T.  (TO  is  said  to 
be  found  by  inter- 
polation.") This 
diagram  has  the 

property     that      the 

distance  traversed  by 
the  body  in  any  in- 
terval T±Tn  is  represented  by  the  area  bounded  by  the 
curve,  the  horizontal  time  line,  and  the  ordinates  at  T^ 
and  Tn.  This  is  evident  from  the  fact  that  the  distance 
traversed  in  any  very  short  time,  T,  is  equal  to  the  speed 
at  the  middle  of  the  time  multiplied  by  T  and  is,  there- 
fore, represented  by  the  area  of  the  narrow  trapezium 
that  stands  on  the  length  representing  T. 

29.  Uniform  Circular  Motion.  —  When  a  point  revolves  in 
a  circle  at  a  constant  rate,  the  magnitude  of  its  velocity, 
that  is,  its  speed,  remains  constant,  but 
the  direction  of  the  velocity  is  constantly 
changing. 

If  s  denote  the  speed,  then  s  is  the 
length  of  the  arc  described  in  unit  time. 
The  angle  through  which  the  radius  turns.  FIG.  la 


VELOCITY  AND  ACCELERATION  31 

in  unit  time  is  s  -r-  r  radians.     This  is  called  the  angular 

velocity  and  is  denoted  by  co.     Hence  o>  =  — 

r 

If  T  is  the  time  required  for  a  revolution,  since  the 
radius  turns  through  2  TT  radians  in  time  T 

27T 


If  n  is  the  frequency  of  the  motion  or  the  number  of 
revolutions  in  unit  time,  nT=  1  and 

«  =  2  TTU. 

30.  Accelerated  Angular  Velocity.  —  When  an  angular  ve- 
locity increases  at  a  constant  rate,  it  is  said  to  be  subject 
to  a  constant  angular  acceleration.  If  we  denote  the  con- 
stant angular  acceleration  by  a  we  may  say  that  a  is  the 
change  of  a>  in  time  t  divided  by  t.  If  o>0  is  the  angular 
velocity  at  the  beginning  of  t  and  o>  that  at  the  end  of  t 

co  =  ft>0  +  at. 

The  mean  angular  velocity  in  this  time  t  is  the  angular 
velocity  at  time  J  t  or  o>0  +  1  at.  Henoe  the  angle  de- 

scribed in  time  t  is 

9  =  Oo  +  £  aO^ 


These  formulae  are  analogous  to  the  formulae  for  accel- 
erated linear  velocity,  angular  displacement  corresponding 
to  linear  displacement,  angular  velocity  to  linear  velocity, 
and  angular  acceleration  to  linear  acceleration  (see  (d) 
p.  37). 

When  a  rigid  body  is  in  rotation  about  an  axis  each 
point  in  the  body  rotates  in  a  circle  whose  centre  is  on 


32  KINEMATICS 

that  axis  and  all  points  necessarily  have  the  same  angular 
velocity  and  the  same  angular  acceleration.  The  angular 
velocity  and  acceleration  of  the  body  are  those  of  any 
particle  in  the  body. 

When  a  point  revolves  in  a  circle  of  radius  r  with  an 
angular  acceleration,  a,  its  linear  speed  along  the  tangent 
to  the  circle  changes  by  ar  in  unit  time  (§  29).  Hence 
if  a  is  its  linear  acceleration  a  =  ar. 

31.  Graphical  Representation  of  Angular  Velocities.  — 
When  a  body  rotates  about  any  axis,  its  angular  velocity 
may  be  represented  in  magnitude  and  direction  by  a  length 
proportional  to  the  magnitude  of  the  velocity,  laid  off  on 
the  axis  of  rotation,  the  direction  of  this  length  being 
related  to  the  direction  of  rotation  as  translation  to  rota- 
P  tion  in  the  motion  of  an 

ordinary  or  "right-handed" 
screw. 

Suppose  LM  to  be  the 
axis  of  rotation  and  AB  to 
be  the  length  laid  off  on  LM 


A  M    to    represent    the    angular 

velocity.  If  p  is  the  per- 
pendicular distance  of  a  particle  P  from  LM,  the  linear 
speed  of  P  is  represented  by  p  -  AB  (§  29)  or  twice  the 
area  of  the  triangle  APB,  and  as  AB*  is  from  left  to 
right  in  the  diagram,  the  linear  speed  of  P  is  toward  the 
reader  and  perpendicular  to  the  plane  APB. 

*  AB  is  in  this  case  a  localized  vector,  since  it  stands  for  quantity  that 
has  magnitude  and  direction  and  also  relates  to  a  definite  straight  line 
LM,  the  axis  of  rotation. 


VELOCITY  AND  ACCELERATION  33 

32.  Composition  of  Angular  Velocities.  —  Let  us  suppose 
that  a  body  has  simultaneously  two  angular  velocities 
about  axes  intersecting  in  J.,  and  let  the  magnitudes  and 
directions  of  these  angular  velocities  be  represented  by 
AB  and  AC  respectively.  The  resultant  of  these  com- 
ponent velocities  is  a  rotation  about  some  axis  through  A. 
We  shall  show  that  the  resultant  is  an  angular  velocity 
about  the  diagonal  AD  of  the 
parallelogram  ABDC  and  is  rep- 
resented by  AD.  Consider  the 
motion  of  a  point  P  in  the  plane 
of  ABDC.  The  two  separate 
linear  speeds  of  P  are  repre- 
sented by  twice  the  areas  of  the 
triangles  APB  and  APQ  respec- 
tively. Now  the  area  of  the  triangles  APD  equals  the 
sum  of  the  areas  of  APB  and  APQ  (these  triangles 
are  on  the  same  base  AP  and  the  distance  of  D  from  AP 
equals  the  sum  of  the  distance  of  B  and  C  from  AP). 
Hence  twice  the  area  of  the  triangle  APD  represents  the 
resultant  linear  speed  of  P.  Thus  P,  and  therefore 
every  other  point  in  the  body,  rotates  about  AD  with 
an  angular  velocity  represented  by  AD. 

Hence,  angular  velocities  about  intersecting  axes  are 
compounded  like  linear  velocities.  The  same  law  evi- 
dently holds  for  angular  accelerations,  since  they  are 
increments  of  angular  velocities  in  unit  time. 

The  reader  should  note  that  the  axes  we  have  been 
speaking  of  are  certain  lines  in  space,  not  certain  lines 
fixed  in  the  moving  body.  The  difference  is  important, 
since  a  line  in  the  body  changes  its  position  as  the  body 


34  KINEMATICS 

rotates.  The  propositions  hold  true,  however,  for  lines 
in  the  body  that  coincide  at  any  moment  with  the  axes  in 
space,  provided  it  be  understood  that  we  limit  ourselves 
to  the  angular  velocities  at  that  instant,  i.e.  to  instantane- 
ous velocities. 

33.  Linear  Acceleration  of  a  Point  that  moves  in  a  Circle 
with  Constant  Speed.  —  When  a  point  revolves  in  a  circle 
with  constant  linear  speed,  although  the  magnitude  of 
the  linear  velocity  is  constant,  the 
direction  of  the  linear  velocity  is 
constantly  changing  and  hence 
there  is  a  linear  acceleration. 
When  the  point  is  at  $,  it  is  mov- 
ing in  the  direction  of  the  tangent 
QT  with  a  speed  s.  Let  us  repre- 

FIG.  16. 

sent  the  velocity  at  that  moment 

by  a  line  oq.  Similarly,  when  the  moving  point  is  at  Q', 
its  velocity  may  be  represented  by  oq' ,  and  oq  and  oq'  will 
be  equal  in  length  since  the  speed  is  constant.  From  the 
triangle  oqqf  we  see  that  the  velocity  which  has  been 
added  to  oq  to  produce  oq'  is  qq' .  If  t  is  the  time  in 
which  the  moving  point  passes  over  the  distance  QQ',  the 
mean  acceleration  in  the  interval  t  is  qq'  -*-t.  If  Q  and  Q' 
be  supposed  indefinitely  close  together,  the  mean  accelera- 
tion in  the  interval  t  will  equal  the  instantaneous  accelera- 
tion at  the  middle  of  t  when  the  moving  point  was  at  P, 
half  way  between  Q  and  Q' .  Hence  if  a  be  the  acceleration 

at  P,  a  =  qq'  -f- 1  and 

qq'  =  at.  (1) 

It  is  readily  seen  from  geometry  that  qq'  is  in  the  direc- 
tion PO,  or  the  acceleration  is  towards  the  centre  of  the  circle. 


VELOCITY  AND  ACCELERATION  35 

Since  QQ'  is  traversed  in  time  t  with  speed  s, 


*t.  (2) 

From  the  similarity  of  the  triangles  OQQf  and  oqqr 


oq  ~   OQ' 

and  since  the  arc  QQ'  is  indefinitely  short,  it  may  be  taken 
as  equal  to  the  chord  QQ'  .  Substituting  in  (3)  from  (1) 

and  (2)  we  get 

at_st 

s  ~~  r 

s2 
Hence  a  =  —  , 

and,  as  we  have  seen,  a  is  towards  the  centre  of  the  circle. 
The  effect  of  this  acceleration  is  to  produce  a  change  in 
the  direction  of  the  velocity  without  affecting  the  magni- 
tude of  the  velocity. 

If  the  angular  velocity  in  the  circle  is  co  and  the  fre- 
quency is  w,  s  =  ft)r=27mr. 

/.  a  =  co2r  =  4  7r2n2r. 

When  a  point  moves  in  a  circle  with  varying  speed,  in 
addition  to  the  acceleration,  a,  towards  the  centre,  there 
must  be  an  acceleration,  a1  ',  along  the  tangent.  The  wThole 
acceleration  will  be  the  resultant  of  these  two  component 
accelerations,  a  and  ar  . 

34.  Acceleration  of  a  Point  moving  in  Any  Curve.  — 
Through  any  point  P  on  a  curve  a  circle  may  be  drawn 
that  coincides  exactly  with  the  curve  at  the  point  P. 
This  circle  is  called  the  circle  of  curvature  of  the  curve  at 
P,  and  its  radius  r  is  called  the  radius  of  curvature  at  P. 


36  KINEMATICS 

A  point  moving  along  the  curve  on  reaching  the  position 

P  is  for  a  moment  moving  in  the  circle  of  curvature  at  P 

s2 

and   hence   has  an  acceleration  --  directed   towards  the 

r 

centre  of  curvature.  If  the  speed  of  the  moving  point 
be  variable,  there  will  also  be  an  acceleration  along  the 
tangent. 

Exercise  IV.    Curve  of  Speed  of  a  Projectile 

This  exercise  is  a  further  study  of  the  curve  of  descent  of  a  falling 
body  obtained  in  Exercise  III. 

Let  the  x  or  y  of  any  point  P  on  the  curve  of  descent  be  obtained 
from  the  curve ;  then  the  time  when  the  ball  was  at  P  can  be  calcu- 
lated from  the  equation  for  the  horizontal  or  that  for  the  vertical 
motion  (§  27).  With  a  knowledge  of  £,  the  component  velocities  at 
P  can  be  calculated  (§  24).  The  resultant  velocity  can  then  be  found 
from  the  components.  The  values  of  the  velocity  as  obtained  in  this 
way  should  be  tabulated  for  five  or  six  points  on  the  curve. 

With  the  values  of  the  speed  and  time  draw  on  cross-section  paper 
a  diagram  of  speed  on  any  convenient  scale.  The  whole  distance  the 
ball  descended  along  the  curve  can  be  obtained  from  the  diagram  of 
speed  by  counting  up  the  number  of  large  and  small  square  units  in 
the  area  of  the  diagram.  This  area  would  equal  the  length  of  the 
curve  of  descent  if  each  unit  of  length  along  the  time  axis  repre- 
sented a  unit  of  time,  and  each  unit  of  length  along  the  speed  axis 
represented  a  unit  of  speed.  But  if  the  former  represent  m  units  of 
time  and  the  latter  n  units  of  speed,  then  each  unit  of  area  represents 
mn  units  of  length  of  the  curve  of  descent,  and  the  area  of  the  dia- 
gram must  be  multiplied  by  mn  to  get  the  length  of  the  curve  of  descent. 

The  length  of  the  curve  of  descent  can  also  be  obtained  by  direct 
measurement.  Fasten  the  sheet  containing  the  curve  on  a  board  by 
half  a  dozen  ordinary  pins  passing  through  points  on  the  curve,  and 
with  a  strip  of  cross-section  paper  stretched  on  edge  close  against  the 
pins,  measure  the  length  of  the  curve.  Then  allow  for  the  scale  to 
which  the  curve  was  drawn. 


VELOCITY  AND  ACCELERATION  37 

DISCUSSION  AND  PROBLEMS 

(a)  Meaning  of  instantaneous  velocity  and  instantaneous  speed. 

(b)  Definition  and  chief  properties  of  curve  of  speed  (why  not 
"  curve  of  velocity  "  ?). 

(c)  Curve  of  speed  of  a  body  falling  in  a  vertical  line.     Formula 
for  distance  traversed  in  time  t. 

(d)  Curve  of  angular  speed  of  a  body  rotating  with  a  constant 
angular  acceleration  (§  30).     Formula  for  angle  described  in  time  t. 

REFERENCES   FOR   CHAPTER  III 

Ames's  "  Text-book  of  General  Physics,"  Chapter  I. 
Watson's  "  Text-book  of  Physics,"  Book  I,  Chapter  V. 
DanielFs  "  Principles  of  Physics,"  Chapters  II,  V. 
Macgregor's  "  Kinematics  and  Dynamics,"  Chapters  I- VI. 


CHAPTER   IV 


PERIODIC   MOTION 

35.  Periodic  Motion.  —  When  a  point  repeats  a  series  of 
movements  in  successive  equal  intervals,  its   motion  is 
called  periodic.     The  vibrations  of  a  pendulum,  of  a  mass 
attached  to  a  spring,  of  a  point  which  has  a  uniform  cir- 
cular motion,   of  a  planet   rotating   about   the   sun,  are 
periodic  motions.     The  time  required  for  each  complete 
repetition  of  the  motion  is  called  the  period  of  the  motion. 
In  some  respects  steady  rotation  in  a  circle  is  the  simplest 
form  of  periodic  motion. 

36.  Phase  and  Epoch  in  Uniform  Circular  Motion.  —  When 
a  point  P  revolves  uniformly  in  a  circle,  the  simplest  way 

of  describing  its  position  at  any 
time  is  by  stating  the  magnitude 
of  the  angle,  POA,  that  the  radius 
through  P  makes  with  some  fixed 
radius  OA,  it  being  understood 
that  POA  is  measured  from  OA 
in  some  definite  direction  of  rota- 
tion, e.g.  counter-clockwise.  The 
angle  POA  is  called  the  phase 
of  P's  motion,  and  is  denoted 

by  <f>.     If  E  be  the  4 position  of  P  at  the  moment  from 
which  time  is  reckoned,  then  EOA,  or  the  phase  of  P's 


PERIODIC  MOTION 


39 


motion  at  zero  time,  is  called  the  epoch  of  P's  motion,  and 
is  denoted  by  e.  If  P's  angular  velocity  be  o>,  then  at 
time  t  POE=wt,  and 

(f)  =  0)t  -f-  €. 

Phase  is  sometimes  measured  by  the  ratio  of  POA  to 
2  TT,  that  is,  by  the  fraction  of  a  period  that  has  elapsed 
since  P  last  passed  through  the  fixed  point  A.  The  epoch 
is  then  measured  by  the  fraction  of  a  period  required  by 
P  to  move  from  A  to  E. 

37.  Simple  Harmonic  Motion.  —  When  a  point  P  rotates 
in  a  circle  with  constant  speed,  the  projection  of  P  on  a 
diameter  moves  backward  and  forward  along  the  diameter, 
completing  a  whole  vi- 
bration in  the  time  in 
which  P  completes  a 
revolution.  The  mo- 
tion of  M,  the  projec- 
tion of  P,  is  called 
Simple  Harmonic  Mo- 

\  tion.  Hence,  simple 
harmonic  motion  is  the 
projection  of  uniform 

!  circular   motion   on   a 
diameter.     The  velocity  and  acceleration   of  M  are  the 
projections  of  the  velocity  and  acceleration  of  P. 

It  will  help  the  reader  to  realize  the  meaning  of  simple 
harmonic  motion  if  he  imagine  himself  looking  at  a  uni- 
form circular  motion  from  a  very  great  distance  in  the 
plane  of  the  motion.  The  only  part  of  the  motion  seen 
, would  be  the  part  transverse  to  the  line  of  sight. 


FIG.  18. 


40  f  KINEMATICS 

38.  Acceleration  in  Simple  Harmonic  Motion. — The  accel- 
eration of  P  is  o)V  along  P  0.  Let  RP  Q  be  parallel  to  and 
in  the  same  direction  as  A! A,  the  diameter  on  which  the 
motion  of  P  is  projected.  Then  the  acceleration  of  M  in 
the  direction  ^OA  is  j»2r  cos  CPQ,  or  —  co2r  cos  PC  A.  If 
CM  (or  ^g  displacement  as  it  is  always  called  in  S.H.M.) 

be  denoted  by  a;,  cosP(7A  =  -.     Hence,  if  u  be  the  accel- 

r 
eratioii  of  Jfin  the  direction  CA, 


r 

=  —  aPx 

A2 
*•)•*• 

Hence  the  acceleration  is  opposite  to  and  proportional  to 
the  displacement,  and  the  period  of  the  motion  is 


39.  Velocity  in  Simple  Harmonic  Motion.  —  The  velocity 
of  P  is  cor  along  the  tangent  PT.  The  velocity  of  M  in 
the  positive  direction  CA  is  the  component  along  OA  of 
the  velocity  of  P.  Denoting  it  by  v, 


v  =  cor  cos 
=  -  cor  cos  TPR 
—  —  cor  sin  PCM 

=  —  cor 


PERIODIC  MOTION  41 

In  this  expression,  T  is  the  period  of  the  S.H.M.,  and 
r  equals  the  greatest  displacement  in  the  S.H.M.,  or  the 
amplitude  of  the  S.  H.  M.  The  ambiguity  of  sign  due  to 
the  square  root  is  removed  by  considering  that  v  must  be 
positive,  as  M.  moves  from  A'  to  A>  and  negative  from  A 
to  A'. 

40.  Circle  of  Reference  of  a  S.  H.  M.  —  From  §38  it  is 
seen  that  S.  H.  M.  is  a  linear  vibration  in  which  the  accel- 
eration is  opposite  to  and  proportional  to  the  displace- 
ment. This  might  have  been,  and  frequently  is,  taken 
as  the  definition  of  S.  H.  M.  It  was  only  for  convenience 
and  brevity  in  deducing  the  properties  of  S.  H.  M.  that 
it  was  defined  as  projection  of  uniform  circular  motion. 
The  reader  must  guard  himself  against  confusing  a  S.  H. 
M.  and  the  uniform  circular  motion  from  which  it  may 
be  regarded  as  projected.  The  circle  described  with  the 
centre  of  a  S.  H.  M.  as  centre  and  the  amplitude  of  the 
S.  H.  M.  as  radius  is  called  the  circle  of  reference  of 
the  S.H.M. 

Any  linear  vibration  that  has  the  characteristic  that 
a  =  —  /AX,  //.  being  a  constant  throughout  the  vibration,  is 
a  S.H.M.,  and  the  period  of  the  vibration  is 


Exercise  V.     Graphical  Study  of  S.H.M. 

Method.  —  A  large  sheet  of  paper  is  tacked  to  the  vertical  cross- 
section  board  used  in  Exercise  III.  We  shall  suppose  that  the  paper 
is  plain ;  but  the  use  of  accurate  cross-section  paper,  with  millimetre 
divisions,  would  simplify  the  work  in  ways  that  will  be  readily  seen. 
On  the  paper  a  quadrant  PA  of  a  circle  is  drawn  with  care  by  means 


42 


KINEMATICS 


of  a  hard  sharp-pointed  pencil.     (The  pencil  may  be  firmly  clamped 
by  means  of  a  screw  pinch-cock  to  a  rod  of  wood,  through  one  end  of 

which  a  sewing  needle 
is  driven.)  Through  the 
centre  of  the  circle  a 
vertical  and  a  horizontal 
radius  are  drawn.  The 
point  on  the  arc  through 
which  k>  draw  the  verti- 
cal radius  should  be 
found  accurately  by 
means  of  a  plumb-line 
of  fine  silk  thread,  the 
bob  being  allowed  to 
hang  in  a  glass  of  water 
to  prevent  oscillations. 
The  direction  of  the 
horizontal  radius  may 
be  found  by  a  metre 


O       m^ 


FIG.  19. 


stick  and  level.  The 
surface  of  the  paper 

should  be  as  nearly  plane  and  vertical  as  possible.  The  board  may 
be  levelled  by  loosening  the  screws  in  the  feet  of  the  supports  and 
placing  thin  wedges  under  them. 

Imagine  that  a  point  P  describes  with  constant  speed  the  circle  of 
which  PA  is  a  quadrant  and  carries  a  plumb-line  that  always  remains 
vertical.  The  intersection  of  the  plumb-line  and  OA  will  have  a 
S.  H.  M.  along  OA,  its  amplitude  being  OA  or  r,  and  its  period  the 
same  as  the  period  T  of  P.  Divide  the  arc  PA  with  great  care  into 
8  or  10  equal  parts,  PPV  P\P&  •••»  and  mark  the  points  of  division  as 
sharply  as  possible.  Pv  P2,  ...  will  be  the  position  of  P  after  succes- 
sive equal  intervals,  each,  say,  of  length  T.  If  Mv  M2,  •••  be  the  pro- 
jections of  Pv  P2,  ...,  they  will  be  the  successive  positions  of  M  at  the 
ends  of  these  equal  intervals  and  the  corresponding  displacements  of 
AT  will  be  OMV  OM2,  .... 

Let  pl  be  the  middle  of  the  arc  PPV  p2  that  of  P^y  and  so  on, 


PERIODIC  MOTION  43 

and  let  nv  n2,  •••  be  the  projections  of  pv  p2,  —.  Then  the  velocities 
of  M  at  the  middle  of  the  successive  equal  time  intervals  will  be 
proportional  to p^nv  p2n2,  •••  (§  39). 

Measurements.  —  To  find  the  displacements  OMV  OM^  ...,  fasten  a 
thin  steel  tape  along  OA  and  note  carefully  its  intersection  with  the 
plumb-line  estimating  to  |  mm.  (a  metre  scale  clamped  to  the  frame- 
work below  the  board  may  be  used  instead  of  the  tape).  The 
plumb-line  should  pass  accurately  through  Pv  Py  •  •«.  The  initial 
displacement,  x0,  of  M  is  0;  xl  =  OMl]  x2  =  OM2',  ....  Tabulate 
these  values  of  x  and  also  the  values  of  pvnv  p2n2,  «... 

Law  of  Velocity.  —  The  distance  traversed  in  the  successive  equal 
intervals  are  dj_  -  x^  -  XQ  ;  d2  =  x2  -  z0;  ....  The  mean  velocities  in 
these  intervals  are  t?x  =  dl  +  T,  v2  =  d2  •+•  T,  «••.  For  T,  any  value,  say 
•j-1^  sec.,  may  be  assumed,  and  T  and  o>  may  then  be  deduced  from  the 
number  of  parts  in  the  quadrant.  The  mean  velocities  in  the  succes- 
sive intervals  may  be  taken  as  the  velocities  at  the  middle  of  the 
intervals,  or  co  'P^v  o>  •  p2ny  •  •-.  Hence  vl  -f- p^n^  v2  -t- p2n2, •  ••  should 
be  nearly  equal  and  their  mean  should  be  o>. 

Law  of  Acceleration.  —  The  increment  of  the  mean  velocity  from 
the  middle  of  the  first  interval  to  the  middle  of  the  second  interval  is 
v2  —  vv  Hence  the  mean  acceleration  in  this  time  is  a1  =  (v2  —  v^  -r-  T, 
and  thus  may  be  taken  as  the  acceleration  at  the  end  of  the  first 
interval,  or  —  w2 .  xv  Similarly  the  acceleration  at  the  end  of  the 
second  interval  is  a2  =  (v3  —  v2)  -=-  r,  and  so  on.  Hence  at  -f-  xv  a2-t-x2, 
•  ..,  should  be  nearly  equal  and  their  mean  should  give  —  o>2. 

Source  of  Error.  —  It  will  be  noticed  that  we  have  spoken  of  mean 
velocities  and  mean  accelerations.  This  is  because  we  could  not 
measure  indefinitely  short  intervals  and  so  get  instantaneous  velocities 
and  accelerations.  It  can,  however,  be  shown  that  if  T  is  only  ^  of 
T,  the  errors  from  this  cause  are  small.  Such  an  error  is  an  error  of 
method.  There  is  a  more  serious  source  of  possible  error  in  this  exer- 
cise. The  first  couple  of  intervals  are  so  nearly  equal  that,  when  we 
take  differences  to  find  the  acceleration,  the  difference  may  be  in  error 
by  a  large  fraction  of  itself,  owing  to  unavoidable  errors  of  measure- 
ment, and  this  may  cause  the  value  of  a±  and  possibly  of  a2  to  be  very 
imperfect.  Such  an  error  is  an  error  of  measurement  or  of  observation. 


44 


KINEMATICS 


DISCUSSION 
(a)  Relation  of  U.  C.  M.  and  S.  H.  M. 

(6)  Velocity  for  various  values  of  x,  e.g.  +  r,  —  r,  0,  +£r,  —  ^r,  etc. 
(c)  Acceleration  for  various  values  of  x. 

(//)  How,  from  the  velocity  or  acceleration  at  a  given  displace- 
ment, could  the  time  required  for  a  whole  vibration  be  calculated  ? 

41.  Phase  and  Epoch  in  S.  H.  M.  —  The  terra  phase  is  used 
in  S.  H.  M.  to  denote  the  position  and  direction  of  the 
motion  of  the  vibrating  point  at  any  time.  The  measure 
of  the  phase  in  the  S.H.  M.  is  taken  as  the  same  as  that 
in  the  uniform  circular  motion  from  which  the  S.  H.  M. 
may  be  supposed  to  be  projected.  When  the  point  which 
has  S.H.M.  is  at  M  (Fig.  18),  its  phase  is  AGP.  The 
epoch  of  the  S.  H.  M.  is  similarly  the  epoch  of  the  circular 
motion,  so  that  we  have,  as  in  uniform  circular  motion, 
<  =  cot  -f-  e 


As  in  uniform  circular  motion  the  phase  and  epoch  of  a 
S.  H.  M.  may  also  be  measured  in  fractions  of  a  period. 

42.  Resolution  of  Uniform  Circular  Motion  into  Two 
S.  H.  M.'s  at  Right  Angles.  —  If  P 
rotates  uniformly  in  a  circle,  and 
if  M  and  N  are  the  projections 
of  P  on  A'  A  and  B'B,  two  di- 
A  ameters  at  right  angles,  then  the 
motions  of  M  and  JVare  S.H.  M.'s, 
and  taken  together  they  make  up 
the  motion  of  P.  These  two 
S.H.  M.'s  have  the  same  ampli- 
tude and  period,  but  when  M  is 


PERIODIC    MOTION  45 

at  its  greatest  displacement,  that  is,  at  A  or  A',  N  is  at 
zero  displacement.  Hence  one  motion  is  one-fourth  of  a 
vibration  ahead  of  the  other,  or  the  motions  differ  in  phase 
by  one-fourth  of  a  period  or  by  %  TT.  This  is  also  the  dif- 
ference of  the  epochs,  since  the  epoch  of  a  S.  H.  M.  is  its 
phase  at  time  t  =  0. 

43.  Trigonometrical  Expression   for   the   Displacement   in 
S.H.  M. — If  r  is  the  amplitude  of  a  S.  H.  M.  and  x  the 

displacement  at  time  £,  it  is  obvious  from  Fig.  18  that 

x  =  r  cos(o>£  +  e). 

If  the  vibrating  point  is  at  its  greatest  positive  displace- 
ment (i.e.  at  A  in  Fig.  18)  when  t  =  0,  then  e  =  0  and 

x  =  r  cos  cot. 
If,  on  the  other  hand,  M  is  at  zero  displacement  when 

t  =  0  and  is  moving  in  the  positive  direction,  then  e  =  —  — 

and 

x  =  r  cos  (cot  —  J£  TT) 

=  r  sin  cot. 

2  7T 

If  we  replace  co  by  — -,  T  being  the  period  of  the  S.H.M., 

we  get  trigonometrical  expression  for  S.  H.  M.  in  which 
no  reference  to  circular  motion  appears. 

44.  The  Simple  Pendulum.  —  A  simple  pendulum  consists 
of  a  small  heavy  body  (usually  a  sphere)  called  the  bob 
suspended  by  a  cord  (or  wire)  whose  weight  may  be  neg- 
lected compared  with  the  weight  of  the  bob.     When  set 
vibrating  through  a  small  angle  the  bob  describes  a  small 
arc  that  is  approximately  a  straight  line. 


46 


KINEMATICS 


Let  6  be  the  angle  that  the  cord  makes  at  any  time 
with  the  vertical.  If  allowed  to  fall  vertically  the  acceler- 
ation of  the  bob  would  be  the  acceleration  of  gravity,  g. 
Since  it  is  confined  to  the  arc  the  acceleration,  a, 

of  the  bob  in  the  positive  direction  of 

its  motion  is 


a  *=  —  g  cos    -  - 


=  —  g  sin  0 


g  /sin  0 
l\    0 

x  being  the  displacement  of  the  bob 
from  the  centre  of  its  path  of  vibra- 

FIG.  21.    •  g       tion.     If  6  be   less   than   3°,   f— 

V    6 

will   differ  from  unity  by  less  than  3  parts  in  10,000. 
Taking  it  as  unity, 


Hence  the  motion  is  (§  40)  S.  H.  M.  and 


45.  Vibrations  of  a  Tuning-fork  and  of  a  Weight  suspended 
by  a  Spring.  —  It  will  be  shown  later  that  a  point  on  a 
tuning-fork  executes  a  motion  that  is  very  nearly  S.  H.  M. 
and  that  the  same  is  true  of  a  mass  suspended  by  a  spiral 
spring  and  vibrating  vertically. 


PERIODIC  MOTION 


47 


Exercise  VI.    Study  of  Motion  of  Pendulum 

A  heavy  iron  cylinder  forms  the  bob  of  a  pendulum,  the  suspension 
being  a  steel  rod  the  mass  of  which  is  small  compared  with  that  of  the 
cylinder.  The  ends  of  the  cylinder  are  parallel  to  the  plane  of  vibra- 


FIG.  22. 


tion,  and  to  one  end  a  glass  plate,  on  which  a  millimetre  scale  is  etched, 
can  be  attached  by  three  small  clips.  A  tuning-fork  is  clamped  to 
the  framework  that  supports  the  pendulum  in  such  a  position  that  its 


48  KINEMATICS 

prongs  vibrate  parallel  to  the  plane  of  vibration  of  the  pendulum. 
A  fine  needle-point  carried  by  a  light  flexible  strip  of  brass  that  is 
attached  to  one  prong  of  the  fork  presses  against  the  glass  plate.  If 
the  plate  be  coated  with  soap  (bon  ami)  and  allowed  to  dry,  the  needle- 
point will  trace  a  clear  sharp-cut  curve  on  the  glass  plate  when  both 
pendulum  and  tuning-fork  are  in  vibration. 

The  pendulum  should  be  drawn  aside  about  5  cm.  and  held  by  a 
cord  which  passes  through  a  couple  of  screw  eyes  (as  in  Fig.  22), 
and  is  held  by  pressure  of  a  thumb  on  the  table ;  when  the  thumb  is 
removed  the  pendulum  will  be  released  without  any  jar.  The  tuning- 
fork  is  started  by  a  blow  from  a  small  wooden  mallet  and  then 
the  pendulum  is  released.  To  prevent  confusion  of  the  record  the 
pendulum  should  be  arrested  at  the  end  of  half  of  a  complete  vibra- 
tion. When  the  tuning-fork  has  stopped  the  pendulum  should  be 
released  and  allowed  to  complete  the  vibration  and  again  arrested. 
The  pendulum  should  then  be  allowed  to  stand  vertical  and  quite  at 
rest  and  the  fork  set  into  a  slight  vibration  so  as  to  give  a  record  of 
the  middle  of  the  arc  of  vibration. 

If  the  plate  be  removed  from  the  pendulum  and  held  up  in  front  of 
a  window  (or  placed  at  an  angle  in  front  of  a  mirror  that  rests  on  the 
table  so  as  to  reflect  light  from  the  window)  and  then  examined 
through  a  magnifying  glass,  the  length  of  the  successive  half-waves  of 
the  curve  can  be  read  on  the  etched  scale  with  considerable  accuracy. 
Assuming  that  each  vibration  of  the  tuning-fork  is  completed  in  the 
same  time,  T,  each  group  of  3  waves  in  the  record  will  be  the  distance  the 
bob  of  the  pendulum  travels  in  time  3  r.  The  lengths  of  the  successive 
groups  should  be  recorded  with  the  greatest  possible  accuracy.  The 
nature  of  the  motion  should  then  be  studied  as  in  Exercise  V,  the 
successive  groups  corresponding  to  OMV  M1M2  -"in  that  exercise. 

From  the  known  frequency  of  vibration  of  the  tuning-fork  and  the 
record  just  obtained  the  period  of  vibration  of  the  pendulum  may  be 
calculated  by  means  of  the  formula  in  §  39.  (See  also  (#)  in  the 
discussion  of  Exercise  Y.)  The  period  of  vibration  should  also  be 
calculated  by  means  of  the  formula  for  the  simple  pendulum,  the 
length  of  the  pendulum  being  taken  as  the  distance  between  the  knife- 
edges  and  the  centre  of  the  bob.  Finally  the  period  should  be  found 


PERIODIC  MOTION  49 

experimentally  by  counting  the  number  of  vibrations  in  two  or  three 
minutes. 

(A  scale  etched  on  the  glass  is  not  indispensable.  If  the  glass  is 
plain,  the  wave  lengths  may  be  read  by  placing  the  glass,  face  down, 
on  a  millimetre  scale.) 

DISCUSSION 

(a)   Sources  of  error. 

(i)  In  what  respect  is  the  motion  of  the  pendulum  not  exactly 
S.H.M? 

(c)  What  form  of  curve  would  be  obtained  if  the  bob  of  the  pendu- 
lum were  replaced  by  a  body  moving  horizontally,  (1)  with  constant 
velocity,  (2)  with  constant  acceleration  ? 

(c?)  What  curve  would  be  obtained  if  the  bob  of  the  pendulum 
were  replaced  by  a  body  falling  freely  while  the  tuning-fork  vibrated 
horizontally? 

(e)  Calculate  the  length  of  a  second's  pendulum. 

(/)  Plow  much  shorter  than  a  second's  pendulum  would  a  clock 
pendulum  be  that  lost  one  minute  per  day? 

(#)  A  pendulum  which  is  a  second's  pendulum  where  g  =  980 
vibrates  3599  in  an  hour  at  the  top  of  a  mountain.  Find  the  accel- 
eration of  gravity  at  that  point. 

46.  The  Projection  of  a  S.  H.  M.  on  a  Straight  Line  in  the 
Same  Plane  is  also  a  S.H.M.  —  Let  Jjf.be  a  point  having 
S.H.M.  along  A  A.  If  the  projections  of  M  and  A  be 
-ZVand  B  respectively,  then  B; 

ON=  OB 
OM     OA 

But    OM=  OA  cos  (at  +  e),  NN 

"D 
. ' .    ON=  OB  COS  (a>t  +  e).  FIG.  23. 

Hence  the  projection  of  a  S.H.M.  is  a  S.H.M.  of  the 
same  period.  By  reversing  the  proof  it  can  be  shown 
that  a  linear  vibration  that  projects  into  a  S.  H.  M.  is  it- 
self a  S.  H.  M. 


50 


KINEMATICS 


47.  Composition  of  S.  H.  M.'s  in  Lines  at  Right  Angles.  —  If 
a  point  has  one  S.  H.  M.  in  a  vertical  line  and  another  in 
a  horizontal  line,  its  resultant  displacement  at  any  time 
can  be  found  by  compounding  its  vertical  displacement, 
y,  and  its  horizontal  displacement,  #,  at  that  time.  These 
might  be  found  from  the  trigonometrical  expressions  for 


A' 


M'  0; 


R' 


B' 
FIG.  24. 


S' 


the  two  S.  H.  M.'s.     They  can  also  be  readily  found  by 
projection  from  the  corresponding  circular  motion. 

(1)  The  simplest  case  is  when  the  S.H.  M.'s  have  the 
same  period,  amplitude,  and  phase.  Then  the  circles  of 
reference  are  coincident.  The  two  S.H.  M.'s  may  be  re- 
garded as  beginning  at  0  at  the  same  time.  The  hori- 
zontal motion  is  the  projection  on  A' A  of  the  motion  of  a 
point  P  that  revolves  in  the  circle  beginning  at  B!,  and 


PERIODIC  MOTION  51 

the  vertical  motion  is  the  projection  on  BfB  of  the  motion 
of  a  point  Q  that  revolves  in  the  circle  beginning  at  A. 
It  is  obvious  that  OL  will  always  be  equal  to  OM  and 
that  the  point  T  that  has  both  of  these  motions  will  al- 
ways lie  in  the  bisector  R1 R  of  the  angles  AOB  and 
A'  OB' .  Moreover,  it  is  clear  that  the  motion  of  M is  the 
projection  of  the  motion  of  T.  Hence  by  §  46  the  motion 
of  T  is  also  a  S.  H.  M.  of  the  same  period  as  the  com- 
ponents. 

(2)  When  the  period  and  amplitude  are  equal  but  the 
phases  differ  by  TT  or  one-half  of  a  vibration,  it  can  be 
shown  in  the  same  way  that  the  resultant  is  a  S.H.M. 
along  the  bisector  8' 8  of  the  angles  B'OA  and  BOA'. 
The  only  difference  in  the  proof  consists  in  supposing  P 
to  begin  at  B  instead  of  at  B1 '. 

(3)  When  the  periods  and  amplitudes  are  equal  and 
the  phases  differ  by  J  TT  or  one-fourth  of  a  period,  the 
motion  is  uniform  circular  motion  in  the  circle  of  refer- 
ence, as  has  been  already  shown  in  §  42. 

(4)  When  the  phase   difference  is  anything  else,  the 
period  and  amplitude  still  being  equal,  the  path  can  be 
shown  to  be  an  ellipse  inscribed  in  JRSR'S'. 

(5)  Let  us  next  suppose  that  the  periods  are  equal  but 
the  amplitudes   different,  the    amplitude  of  the  vertical 
motion  being  the  greater.     The  result  will  be  a  relative 
elongation  of  all  the  vertical  lines  of  Fig.  24.     Corre- 
sponding to  (1)  and  (2)  above  we  shall  still  have  motion 
in  straight  lines  but  the  lines  will  be  closer  to  the  vertical, 
and  corresponding  to  (3)  and  (4)  we  shall  have  motion 
in  ellipses  whose  directions  depend  on  the  phase  relations. 

(6)  If  the  periods  be  slightly  different,  then  one  motion 


52 


KINEMATICS 


\6 


will  continually  gain  in  phase  on  the  other ;  at  any  moment 
the  motion  will  be  in  an  ellipse  whose  form  and  position 
will  depend  on  the  difference  of  phase  at  that  moment, 
but  the  ellipse  will  be  continually  changing  in  form  and 
position,  passing  through  the  circle  ,and  straight  line  as 
particular  cases. 

48.  Composition  of  S.  H.  M.'s  of  Different  Periods  in  Lines  at 
Right  Angles.  —  Let  P  have  a  vertical  S.  H.  M.,  of  which  the 
amplitude  is  4  cm.,  represented  by  OA  and  the  period 
3  sec.,  and  a  horizontal  A  n 

S.H.M.,  of  which  the 
amplitude  is  2  cm.,  rep- 
resented by  OB  and  the 
period  2  sec.  To  avoid 
confusion  draw  the  cir- 
cle of  reference,  (7,  of 
the  vertical  S.  H.  M.  at 
a  distance  from  0  with 
its  centre  on  a  horizontal 
line  through  0  and  the 
circle  of  reference,  C',  of 
the  horizontal  S.H.  M. 
at  a  distance  from  0  with  its  centre  on  a* vertical  line 
through  0.  If  O  be  divided  into  twelve  equal  parts 
and  Or  into  eight  equal  parts,  then  the  time  of  de- 
scribing each  part  of  0  and  O'  will  be  the  same,  namely 
^  sec.  Consider  the  case  in  which  the  vibrating  point 
that  has  the  sum  of  the  two  S.H.M.'s  is  at  its  greatest 
displacement,  OA,  in  the  vertical  S.H.M.  when  it  is  at 
its  greatest  displacement,  OB,  in  the  horizontal  S.  H.  M. 


\ 


FIG.  25. 


PERIODIC  MOTION  53 

Then  at  time  t  =  0  the  resultant  position  is  the  intersec- 
tion of  a  horizontal  line  through  point  1  of  circle  0  and 
a  vertical  line  through  point  1  of  circle  O' .  At  time 
t  =  \  sec.  the  resultant  position  is  the  intersection  of  a 
horizontal  line  through  point  2  of  (7,  and  a  vertical  line 
through  point  2  of  C',  and  so  on.  Thus  the  curve  in 
which  the  vibrating  point  moves  can  be  traced  out  point 
by  point. 

If  the  phase  relation  between  the  two  S.H.M.'s  be 
something  different  from  the  above,  a  different  resultant 
curve  will  be  obtained.  The  different  curves  obtained  by 
making  the  vertical  motion  start  -^  sec.,  ^  sec.,  |  sec.,  etc., 
later  than  the  horizontal  motion  are  readily  drawn. 

If  the  ratio  of  the  periods  be  slightly  different  from 
3 :  2,  the  resultant  curve  will  pass  continuously  through 
each  of  these  forms  and  back  again. 

For  other  ratios  of  the  period  similar  sets  of  curves  are 
obtained,  each  set  being  characteristic  of  a  particular  ratio 
of  the  periods. 

Exercise  VII.    Resultant  of  S.H.M.'s  at  Right  Angles 

A  Blackburn  pendulum  is  used  in  this  experiment.  An  endless 
braided  cord  passes  over  two  hooks  in  the  ceiling  and  is  brought 
together  by  a  small  ring  attached  to  one  side  of  the  cord,  so  that 
the  whole  suspension  has  the  form  of  a  Y.  The  bob  is  a  heavy  ring 
of  lead  soldered  to  a  metallic  disk,  and  carrying  a  quantity  of  fine 
sand,  which  can  stream  out  through  a  small  hole  in  the  centre  of 
the  disk.  When  allowed  to  swing,  the  bob  is  subject  to  two 
S.H.M.'s  at  right  angles,  the  period  of  one  being  invariable  and 
determined  by  the  distance  of  the  bob  from  the  ceiling,  while  the 
position  of  the  other  depends  on1  the  position  of  the  small  ring 
through  which  the  cord  passes.  The  motion  of  the  bob  is  traced 


54  KINEMATICS 

on  a  cross-section  board  by  the  stream  of  sand.  The  sand  should  be 
sifted  clean  and  dried  by  being  heated  in  a  dipper  before  it  is  used. 

The  position  of  the  ring  should  first  be  adjusted  so  that  the 
periods  are  in  some  simple  ratio,  say  3  : 4.  This  will  require  a  few 
preliminary  trials,  which,  for  convenience  in  collecting  the  sand, 
may  best  be  made  on  a  sheet  of  paper  placed  below  the  pendulum. 
When  the  proper  adjustment  has  been  attained  so'  that  the  same 
curve  is  traced  over  and  over  again,  the  paper  may  be  replaced  by 
the  cross-section  board  placed  centrally  below  the  pendulum  when 
at  rest.  To  obtain  wide  amplitudes  and  a  clear  curve,  release  the 
bob  a  couple  of  inches  from  the  corner  of  the  board,  and  arrest  the 
motion  when  the  curve  has  been  completed  once.  Then  reproduce 
the  curve  on  cross-section  paper  as  in  Exercise  III,  leaving  the 
adjustment  in  the  meantime  undisturbed. 

The  same  resultant  curve  should  also  be  derived  by  graphical 
composition,  as  described  in  §  48.  Since  the  pendulum  started  from 
rest,  the  phases  of  the  components  were  initially  the  same. 

Before  the  adjustment  is  disturbed  the  effect  of  varying  the  phase 
relation  should  be  studied.  For  this  purpose  the  pendulum  may  be 
started  with  an  impulse  from  various  parts  of  the  board.  Two  or 
three  such  variations  should  be  obtained  and  drawn  free-hand  on 
the  cross-section  paper. 

The  effect  of  slightly  varying  the  ratio  of  the  periods  may  be 
studied  by  slightly  raising  or  lowering  the  small  ring.  The  exact 
ratio  of  the  periods,  if  they  are  commensurate,  may  be  found  by 
counting  the  number  of  oscillations  in  the  two  directions.  Some 
of  the  curves  corresponding  to  different  ratios  of  periods,  such  as 
2:1,  5:3,  etc.,  should  be  obtained  and  drawn  free-hand,  and  the 
effect  of  varying  phase  and  ratio  of  period  examined  as  before. 

Finally  the  ring  should  be  drawn  as  near  to  the  top  as  possible, 
and  the  resultant  of  two  S.  H.  M.'s  of  nearly  the  same  period  studied. 

DISCUSSION 

(a)  Points  on  the  first  curve  drawn  at  which  the  phase  difference 
is  0,  £TT,  TT,  |TT. 

(I)  Does  the  escape  of  sand  change  the  period  ? 


PERIODIC  MOTION  55 

(c)  Ratio  of  the  periods  when  the  ring  is   halfway  between  the 
ceiling  and  the  bob  of  the  pendulum. 

(d)  Where  and  in  what  direction  might  an  impulse  be  given  to 
the  pendulum  without  influencing  the  form  of  the  curve  ? 

49.  Composition  of  S.H.M.'s  in  the  Same  Line. —  Cases  in 
which  a  point  has  two  or  more  S.  H.  M.'s  in  the  same  line 
occur  in  the  transmission  of  sound,  light,  and  electrical 
waves.  Each  component  S.  H.  M.  may  be  regarded  as 
the  projection  of  a  uniform  circular  motion.  Let  us 
suppose  that  P  rotates  steadily  in 
a  circle  and  that  the  motion  of  the 
projection,  L,  of  P  is  one  of  the 
component  S.  H.  M.'s.  Let  another 
point,  Q,  rotate  uniformly  in  an- 
other concentric  circle,  and  let  the 
motion  of  the  projection,  M,  of  Q  be  FlG-  26> 

the  other  component  S.  H.  M.  If  ON  be  always  equal 
to  the  sum  of  OL  and  OM,  the  motion  of  N  will  be  the 
sum  of  the  two  component  S.  H.  M.'s.  ON  is  evidently 
the  projection  of  OR,  the  diagonal  of  a  parallelogram  of 
which  OP  and  OQ  are  adjacent  sides. 

(1)  Let  the  periods  of  the  component  S.  H.  M.'s  be 
equal.  In  this  case  P  and  Q  move  with  equal  angular 
velocities  and  the  angle  POQ  remains  constant.  Hence 
R  rotates  in  a  circle  in  the  same  period  as  P  and  Q,  and 
the  motion  of  N  is,  therefore,  a  S.  H.  M.  of  the  same 
period  as  the  components.  The  phase  of  -ZV's  motion  at 
any  time  is  the  angle  R  ON,  and  this  is  always  intermediate 
between  the  phases  of  the  components  or  the  angles  P  OL 
and  QOM.  The  amplitude,  OR,  of  the  resultant  depends 
on  the  amplitudes,  OP  and  OQ,  of  the  components  and 


56  KINEMATICS 

also  on  the  constant  phase  difference  POQ.  When  POQ 
is  0,  the  components  are  in  the  same  phase,  and  OR  is  the 
sum  of  OP  and  OQ  ;  and  when  POQ  is  180°,  OR  is  the 
difference  of  OP  and  OQ. 

(2)  Let  the  periods  of  the  components  be  different. 
Let  the  period  of  L  and  P  be  Tv  and  let  that  of  M  and  Q 
be  Ty  and  suppose  T±  <  T^  so  that  the  angular  velocity  of 
P  is  greater  than  that  of  Q.  In  this  case,  as  P  and  Q 
rotate,  the  angle  POQ  increases  at  a  constant  rate.  Hence 
R  does  not  move  in  a  circle  and  the  motion  of  N  is  there- 
fore not  a  S.  H.  M.  If  Tl  and  T2  are  nearly  equal,  the 
amplitude  of  the  motion  of  N  varies  between  the  sum  of 
OP  and  PQ  and  their  difference.  The  farther  2\  and  T2 
are  from  equality  the  more  rapidly  do  the  variations  occur. 
The  rate  of  increase  of  POQ  is  the  difference  between  the 
angular  velocities  of  P  and  §,  that  is,  —  —  --  —  •  When 

J-l  ™2 

POQ  is  zero,  the  components  are  in  the  same  phase,  and 
when  POQ  has  increased  to  2  TT,  they  are  again  in  the  same 
phase  ;  if  this  requires  a  time  jP,  the  rate  of  increase  of 
POQ  is  ~>  Hence 

l-.JL     JL 
T     T±      Tj 

or,  denoting  the  respective  frequencies  (§  29)  of  the  com- 
ponents by  Wj  and  n2  and  the  frequency  of  coincidences 


Exercise  VIII.    Resultant  of  S.H.  M.'s  in  the  Same  Line 

Two  light  spiral  springs,  each  carrying  a  weight,  are  placed  some 
distance  apart  so  that  the  weights  are  at  the  same  level.  A  very  light 
wooden  rod  is  suspended  from  the  weights  by  short  threads  that  leave 


PERIODIC  MOTION  57 

the  rod  considerable  freedom  of  motion.     If  one  weight  be  kept  at 
rest  while  the  other  is  in  vertical  vibration,  the  centre,  C,  of  the  rod 
will  have  a  S.  H.  M.  of  the  same  period  as  the  vibrat- 
^  ^     ing  weight  but  of  half  the  amplitude.     When  both 

weights  are  in  vibration  the  motion  of  C  is  the  sum 
of  two  S.  H.  M.'s  in  the  same  line.  A  vertical  scale 
placed  behind  C  will  show  the  amplitudes  of  the 
vibrations. 

If  both  weights  be  drawn  downwards  until  the 
C         y    rod  is  horizontal  and  then  released  simultaneously, 

they  will  start  in  the  same  phase  of  vibration  (that 
FIG.  27. 

of  greatest  displacement).  When  the  rod  is  hori- 
zontal and  for  a  moment  moving  parallel  to  itself,  the  motions  will 
be  again  in  the  same  phase  and  the  time  between  these  two  coin- 
cidences will  be  the  "  coincidence  period,"  T.  To  find  T  by  obser- 
vation, release  the  weights  on  a  tick  of  the  clock  at  the  beginning  of 
a  minute  and  find  the  number  of  minutes  and  seconds  required  for 
several  coincidences.  This  should  be  done  (1)  with  such  weights  and 
springs  that  7\  and  T2  are  quite  different  (e.g.  about  3:2).  (2)  When 
Tl  and  T2  are  not  far  from  equal.  In  the  first  case  the  coincidences 
will  occur  frequently  and  a  large  number  should  be  counted  and 
timed. 

In  each  case  7\  and  T2  should  be  found  by  observing  several  times 
the  number  of  vibrations  in  three  or  four  minutes  and  taking  the  mean. 
From  these  T  should  be  calculated  (§  49).  7\  and  Tz  will  need  to  be 
observed  with  special  care  in  the  second  case  above,  since  their  differ- 
ence is  used  in  calculation. 

Using  a  scale  pan  and  weights  for  one  suspended  body,  adjust  the 
weights  in  the  pan  until  Tl  and  T2  are  equal,  and  then  verify  the 
statements  of  §  49  as  regards  the  composition  of  S.  H.  M.'s  of  the  same 
period ;  vary  both  the  phase  difference  and  the  amplitudes  of  the 
components,  and  note  the  amplitude  of  the  resultants. 

DISCUSSION 

(a)  What  is  the  resultant  of  three  or  more  S.  H.  M.'s  of  the  same 
period  in  the  same  line  ? 


58  KINEMATICS 

(&)  If  two  S.  H.  M.'s  of  the  same  period  in  the  same  line  differ  in 
phase  by  £  of  a  period  (or  90°),  what  is  the  amplitude  of  the 
resultant  ? 

(c)  Find  an  expression  for  the  amplitude  of  the  resultant  of  two 
S.  H.  M.'s  of  the  same  period  and  with  a  phase  difference  of  0. 

(d)  When  the  components  in  (c)  are  of  different  amplitudes,  to 
which  is  the  resultant  nearer  in  phase  ? 

(e)  If  the  coincidence  period  of  two  S.  H.  M.'s  is  observed  and  the 
period  of  one  of  them  is  known,  how  can  that  of  the  other  be  found 
("  coincidence  method  "  of  rating  a  pendulum)  ? 

REFERENCES  FOR  CHAPTER  IV 

Watson's  "  Text-book  of  Physics,"  Book  I,  Chapter  VII. 

Daniell's  "  Principles  of  Physics,"  Chapter  Y. 

Clifford's  "  Kinematic,"  Chapter  I. 

Macgregor's  "  Kinematics  and  Dynamics,"  Part  I,  Chapter  IY. 


DYNAMICS 

CHAPTER  V 
FORCE 

50.  So  far  we  have  considered  the  motion  of  bodies 
which  have  certain  velocities  and  accelerations  without 
inquiring  how  these  velocities  and  accelerations  are  pro- 
duced.    Clear,  systematic  views  as  to  the  way  in  which 
one  body  may  influence  the  motion  of  another  body  were 
first  arrived  at  by  Sir  Isaac  Newton.     His  three  Laws  of 
Motion  lead  to  results  that  have  been  verified  in  innumer- 
able cases  and  this  is  the  ground  of  our  belief  that  they 
are  accurate.       When  once  stated  and  understood  they 
seem  so  nearly  obvious  that  they  are  sometimes  called 
axioms.     Improvements  in  the  way  of  arranging  and  stat- 
ing them  will  no  doubt  come  in  course  of  time,  but  they 
form  at  the  present  time  the  most  convenient  and  satis- 
factory basis  of  Dynamics. 

51.  Newton's  First  Law  of  Motion.  —  "  Every  body  con- 
tinues in  its  state  of  rest  or  of  uniform  motion  in  a  straight 
line  except  in  so  far  as  it  is  compelled  by  external  force  to 
change  that  state." 

Our  primary  conception  of  force  is  a  certain  muscular 
sensation  associated  with  any  attempt  to  move  a  body  or 
change  its  motion.  When  we  see  the  same  effect  produced 
in  some  other  way,  for  example  by  the  impact  of  a  second 

59 


60  DYNAMICS 

body  on  the  first,  we  attribute  the  result  to  a  force  exerted 
by  the  second  body  on  the  first.  This  is  a  somewhat  arti- 
ficial but  very  convenient  extension  of  our  primary  con- 
ception of  force.  A  definition  of  force  in  this  sense  is 
implied  in  Newton's  First  Law. 

Force  is  any  action  between  two  bodies  that  changes  the 
motion  of  either.  This  must  be  understood  as  a  definition 
of  the  meaning  of  force  ;  a  definition  of  the  measure  of  force 
follows  from  Newton's  Second  Law.  When  a  body  at  rest 
begins  to  move  or  when  its  motion  varies  in  either  magni- 
tude or  direction,  the  effect  can  be  traced  to  some  other 
body  which  is  said  to  exert  a  force  on  it.  The  property 
in  virtue  of  which  a  body  not  acted  on  by  any  force  re- 
mains at  rest  or  in  uniform  motion  is  called  inertia. 

52.  Newton's  Second  Law  consists  essentially  of  two 
statements:  (1)  'the  ratio  of  two  different  forces  is  the 
ratio  of  the  accelerations  they  produce  in  the  same  body  ; 
(2)  the  ratio  of  the  masses  of  two  different  bodies  is  the 
inverse  ratio  of  the  accelerations  produced  in  them  by  a 
certain  force.  The  first  enables  us  to  compare  and  meas- 
ure forces ;  the  second  enables  us  to  compare  and  measure 
masses ;  both  are  usually  combined  in  a  single  statement. 
In  giving  it  we  shall  use  two  terms  that  have  become 
current  since  Newton's  time.  The  product  of  a  force  by 
the  time  it  acts  is  called  the  impulse  of  the  force.  The 
product  of  the  mass  of  a  body  by  its  velocity  is  called  the 
momentum  of  the  body. 

"  Change  of  momentum  is  proportional  to  the  impulse 
of  the  force  applied  and  takes  place  in  the  direction  of  the 
force."  If  a  force,  F,  act  for  a  short  time,  £,  on  a  body  of 


FORCE  61 

mass,  w,   and  if  the  velocity  of  the  body  change  from 

vio  v' 


or 

or  Fccma, 

a  being  the  acceleration  of  the  body. 

The  law  refers  only  to  the  change  of  momentum  pro- 
duced by  a  force  and  therefore  implies  that  the  change  of 
momentum  is  the  same  no  matter  what  the  initial  motion 
of  the  body.  It  also  implies  that  the  change  of  momentum 
produced  by  a  force  is  independent  of  the  action  of  other 
forces,  or  when  a  number  of  forces  act  on  a  body  we  may 
calculate  their  results  independently  and  then  compound 
these  results. 

53.  Units  of  Mass  and  Force.  —  Newton's  Second  Law 
may  be  stated  thus  :  ]?  _  £  .  ma 

k  being  a  constant  the  value  of  which  depends  on  the  units 
of  mass  and  force  adopted  (it  being  understood  that  the 
unit  of  acceleration  is  fixed  by  the  units  of  length  and 
time  already  chosen).  If  suitable  units  of  mass  and  force 
be  chosen,  k  will  be  unity.  Let  the  unit  of  mass  be  chosen 
first  and  then  let  the  unit  offeree  be  taken  as  that  force 
which  acting  on  unit  mass  gives  it  unit  acceleration.  Since 
in  this  case  F,  m,  and  a  are  all  unity  at  the  same  time,  k 
must  also  be  unity  and  therefore 

F=  ma. 

The  unit  of  mass  that  is  usually  employed  in  Physics  is 
the  mass  of  a  thousandth  part  of  a  certain  block  of  platinum- 
iridium  kept  at  Sevres  near  Paris  and  known  as  the  kilo- 
gramme prototype.  The  thousandth  part  of  this  mass  is 


62  DYNAMICS 

called  the  gramme.  The  corresponding  unit  of  force,  or  the 
force  that  would  give  a  gramme  an  acceleration  of  1  cm. 
per  second  per  second  is  called  the  dyne.  The  gramme 
and  the  dyne  are  the  units  of  mass  and  force  respectively 
in  the  absolute  C.  G.  S.  (centimetre-gramme-second)  sys- 
tem of  units.  The  gramme  is  (very  nearly)  the  mass  of 
1  cc.  of  water  at  4°  centigrade. 

54.  Mass  and  Weight.  —  Every  body  on  the  surface  of 
the  earth  is  attracted  by  the  earth  with  a  certain  force 
called  the  weight  of  the  body.  Newton  showed  that  at 
any  one  place  the  attractions  on  different  bodies  are  pro- 
portional to  the  masses  of  the  bodies.  The  experiments  by 
which  he  proved  this  consisted  in  timing  pendulums  of  the 
same  length  but  with  bobs  of  different  sizes  and  different 
materials.  He  found  that  they  all  vibrated  in  the  same 
time.  Two  such  pendulums  when  at  the  same  inclination 
to  the  vertical  are  acted  on  in  the  direction  of  motion  by 
the  same  fraction  of  the  force  of  gravity.  But  since  they 
vibrate  in  equal  times  they  must  at  equal  inclinations  to  the 
vertical  have  the  same  acceleration  in  the  direction  of  mo- 
tion. Therefore  the  ratio  that  the  force  in  the  direction  of 
motion  bears  to  the  mass  must,  by  Newton's  Second  Law,  be 
the  same  for  the  two  pendulums.  Hence  the  whole  forces 
of  gravity  on  the  bobs  must  be  proportional  to  their  masses, 
or  weight  is  proportional  to  mass.  This  is  the  basis  of  the 
most  convenient  method  of  comparing  masses,  namely,  by 
comparing  the  weights  of  the  masses  by  means  of  a  balance. 

While  there  is  this  close  connection  between  mass  and 
weight,  it  must  not  be  forgotten  that  mass  or  inertia  is 
essentially  different  from  weight  or  the  force  of  gravity. 


FORCE  63 

At  a  very  great  distance  from  other  attracting  bodies  the 
weight  of  a  body  would  be  very  small,  while  its  mass  is 
everywhere  the  same. 

If  m  be  the  mass  of  a  body  and  IF  its  weight  in  absolute 
units  of  force,  w= 

g  being  the  acceleration  of  gravity,  which  is  the  same  for 
all  bodies  at  the  same  part  of  the  earth. 

55.  Gravitational  Unit  of  Force.  —  In  dealing  with  prob- 
lems in  which  weight  is  the  chief  force  to  be  considered, 
engineers  find  it  convenient  to  use  the  weight  of  a  pound 
(in  English-speaking  countries)  as  their  unit  of  force,  the 
mass  of  a  pound  being  the  unit  of  mass.  In  this  case  the 
value  of  k  in  Newton's  Second  Law  cannot  be  unity.  For, 
if  the  unit  of  force  (a  pound  weight)  act  freely  on  the 
unit  of  mass  (a  pound),  the  acceleration  produced  is  g. 

.-.  1  =  &  •  1  .  #,  or  k  =  — 
y 

Hence,  in  this   case,  the  formula  for  Newton's  Second 

Law  is  1 

F  =  -  ma, 
9 

it  being  understood  that  F  is  measured  in  the  weight  of  a 
pound  as  unit  of  force,  and  m  in  the  pound  as  unit  of 

mass.     The  inconvenient  factor  -  may  be  omitted  if  a 

.  9 
mass  of  ^-pounds  be  taken  as  unit  of  mass. 

Exercise  IX.    Force  and  Acceleration 

Apparatus.  —  A  bicycle  wheel  is  mounted  on  top  of  a  tall  post  and 
a  cord  carrying  large  iron  masses  is  stretched  over  the  wheel.  In  the 
wood  of  the  rim  (and  somewhat  closer  to  one  side  to  avoid  the  spokes) 


DYNAMICS 


a  V-shaped  groove  is  turned,  and  the  cord  rests  in  this  groove.     A 
simple  form  of  clamp  fixed  to  the  post  enables  the  operator  to  keep 

the  wheel  fixed ;  a  slight  jerk  on  a  cord 
attached  to  the  clamp  will  release  the 
wheel.  If  the  axis  of  the  wheel  passes 
through  the  centre  of  gravity  of  the 
wheel  and  also  through  the  centre  of 
the  groove  in  which  the  cord  rests,  the 
wheel  will  have  no  tendency  to  move 
when  the  masses  at  the  end  of  the  cord 
are  equal.  A  small  weight  placed  on  one 
of  the  large  masses  will  cause  the  latter 
to  move  with  a  constant  acceleration 
when  the  wheel  is  released.  Additional 
large  masses  and  different  small  weights 
are  provided  so  that  the  masses  moved 
may  be  changed  and  also  the  forces  caus- 
ing the  motion  varied.  The  observer  can 
note  the  position  of  the  large  masses  at 
any  time  by  reading  a  vertical  scale  which 
stands  close  behind  one  of  them.  The 
cord  sustaining  the  masses  extends  to  the 
floor  on  both  sides,  so  that  its  weight  on 
each  side  is  always  the  same. 

Adjustments.  —  The  supporting  post 
must  be  adjusted  until  the  cord  has  no 
tendency  to  move  out  of  its  groove.  On 
four  of  the  spokes  of  the  wheel  there  are 
small  movable  weights  which  must  be 
adjusted  until  the  wheel  remains  at  rest 
in  any  position  when  the  masses  sup- 
ported by  the  cord  are  equal. 

Distance  and  Acceleration.  — -  When  the 
acceleration  in  the  line  of  motion  is  con- 
stant, the  distance  traversed  from  rest  is  proportional  to  the  square 
of  the  time.     This  may  be  tested  by  attaching  equal  masses   to 


FORCE  65 

the  cord  (preferably  the  largest  masses  supplied)  and  placing  a  small 
weight  on  one.  One  mass  being  elevated  to  a  position  observed 
on  the  vertical  scale  and  the  other  being  near  the  floor,  the  wheel  is 
released  on  a  tick  of  the  clock  and  the  position  of  the  descending  mass 
at  the  third  succeeding  tick  noted  (the  intermediate  seconds  being 
passed  over,  since  the  distances  are  too  small  to  be  accurately  ob- 
served). This  should  be  repeated  twice  and  the  mean  of  the  three 
readings  taken.  The  same  should  be  done  for  each  succeeding  second, 
until  the  descending  mass  reaches  the  floor.  Twice  the  distance  in 
each  case  divided  by  the  square  of  the  time  should  give  a  constant, 
namely,  the  acceleration. 

Device  for  Recording  Distances.  —  In  the  above,  distances  of  descent 
are  supposed  to  be  observed  by  eye.  This  has  the  advantage  of  sim- 
plicity, and  with  care  gives  satisfactory  results.  If  preferred,  a  record 
of  the  motion  may  be  obtained  by  the  following  device.  A  heavy, 
adjustable  pendulum,  adjusted  to  beat  seconds,  swings  from  a  knife- 
edge  attached  to  the  post.  The  rod  of  the  pendulum  extends  above 
the  knife-edge,  and  to  the  upper  end  of  the  extension  a  small  camel's- 
hair  brush  is  attached.  The  brush  is  inked  once  in  a  vibration  by 
touching  a  wet  stick  of  India  ink,  and  as  the  pendulum  passes  through 
the  vertical,  the  brush  makes  a  trace  on  a  strip  of  mucilaged  paper 
wrapped  around  one  side  of  the  rim  of  the  wheel.  When  the  pendu- 
lum is  released  at  the  beginning  of  an  experiment,  it  releases  the  wheel 
on  first  passing  through  the  vertical  and  at  the  same  time  makes  the 
first  record  on  the  paper. 

Since  the  movement  of  the  paper  keeps  pace  with  the  movements  of 
the  weights,  the  record  on  the  paper  is  a  record  of  the  movements  of 
the  weights.  The  paper  is  removed  at  the  end  of  an  experiment  and 
its  record  interpreted. 

Newton's  Second  Law  of  Motion.  —  (1)  The  accelerations  given  to 
a  mass  are  proportional  to  the  forces  applied.  This  may  be  tested  by 
comparing  the  acceleration  just  measured  with  the  acceleration  when 
the  weight  placed  on  the  descending  mass  is  doubled.  All  of  the 
readings  need  not  be  repeated;  it  will  be  sufficient  if  the  distance  for 
one  particular  number  of  seconds  is  determined,  preferably  the  largest 
number  for  which  observations  can  be  conveniently  made. 


66  DYNAMICS 

(2)  The  accelerations  produced  by  a  given  force  are  inversely  as 
the  masses  set  in  motion.  This  may  be  tested  by  replacing  the 
masses  attached  to  the  cord  by  others  half  as  great  and  finding  the 
acceleration  as  before. 

Calculation  of  Distances.  —  From  the  masses,  the  value  of  g  and  the 
greatest  time  observed  in  each  of  the  preceding  cases,  the  correspond- 
ing distances  should  be  calculated.  Any  discrepancy  between  the 
observed  and  the  calculated  distances  should  be  accounted  for. 

Tension  of  Cord.  —  The  equation  for  the  motion  of  each  mass 
should  be  stated  and  the  tension  of  the  cord  deduced  in  each  of  the 
three  cases  studied. 

(This  exercise  will  be  continued  from  the  point  of  view  of  energy 
and  angular  motion  of  the  wheel  in  Exercise  XXV.) 

DISCUSSION  AND  PROBLEMS 

(a)  Meaning  and  deduction  of  formulae  used. 

(&)    Is  the  tension  the  same  in  all  parts  of  the  cord  ? 

(c)  Effect  of  friction. 

(d)  Calculate  what  addition  to  the  large  masses  would  be  equiva- 
lent to  the  mass  of  the  wheel. 

(e)  Calculate  for  one  of  the  cases  studied  the  height  to  which  the 
ascending  mass  rises  after  the  other  mass  strikes  the  floor. 

(/)  Suppose  that  in  one  of  the  cases  studied  the  cord  were  to 
break  when  the  masses  are  at  the  same  level.  (1)  What  would  be 
the  interval  between  the  impacts  of  the  masses  on  the  floor? 
(2)  With  what  velocities  would  they  strike  the  floor? 

(g)  Twelve  bullets  are  divided  between  two  scale  pans  connected 
by  a  cord  passing  over  a  very  light  pulley.  What  division  of  the 
bullets  will  produce  the  greatest  tension  of  the  support  of  the 
pulley  ? 

(A)  A  cord  passes  over  two  fixed  pulleys  and  through  a  third  pul- 
ley suspended  between  them.  A  mass  of  10  kg.  is  attached  to  one 
end  of  the  cord,  a  mass  of  5  kg.  to  the  other  end,  and  the  suspended 
pulley  and  an  attached  weight  weigh  2  kg.  The  parts  of  the  cord 
being  vertical,  with  what  acceleration  will  the  masses  move  if 
released  ? 


FOBCE  67 

56.  Force  of  Gravitation.  —  The  weight  of  a  body  is  a 
particular  case  of  the  attraction  between  bodies  called 
gravitation.  From  a  study  of  the  motions  of  the  moon 
and  the  planets,  Newton  discovered  that  the  accelerations 
of  these  bodies  are  due  to  the  fact  that  between  any  par- 
ticle of  mass  ml  and  another  particle  of  mass  m^  at  a 
distance  r  from  the  first,  there  is  an  attraction  expressed 
by  the  formula 


Gr  being  a  constant  called  the  constant  of  gravitation.  So 
far  as  known  this  law  is  perfectly  exact.  It  is  true  for 
bodies  like  the  planets  and  the  sun  at  great  distances 
apart,  and  very  careful  experiments  have  shown  that  it 
is  true  for  small  bodies  only  a  few  centimetres  apart. 
Whether  it  also  holds  true  for  much  smaller  distances  is 
not  yet  known.  Experiments  have  shown  that  the  attrac- 
tion between  two  bodies  does  not  depend  on  the  materials 
of  which  they  consist  and  is  not  influenced  by  intervening 
bodies. 

Newton  also  showed  that  a  body  of  spherical  shape,  and 
of  the  same  density  at  all  points  equally  distant  from  the 
centre,  attracts  external  bodies  as  if  it  were  concentrated 
at  the  centre.  The  earth  is  very  nearly  such  a  body,  and 
the  attraction  between  it  and  a  body  outside  of  its  surface 
varies  nearly  inversely  as  the  square  of  the  distance  of 
the  body  from  the  centre  of  the  earth.  But  the  earth  is 
not  quite  spherical,  and  a  body  on  the  surface  of  the  earth 
is  farther  from  the  centre  the  nearer  it  is  to  the  equator. 
This  slightly  affects  the  acceleration,  g,  of  a  falling  body, 
and  there  is  also  an  effect  due  to  the  rotation  of  the  earth 


68  DYNAMICS 

(§  64)  and  different  in  different  latitudes.     Measurements 

of  g  by  the  pendulum  agree  fairly  closely  with  the  formula 

g  =  £0(i  -  .0026  cos  2  X  -  .0000003  Z), 

where  A,  is  the  latitude,  I  the  height  above  sea-level  in 
metres,  and  gQ  =  980.6. 

The  value  of  the  constant  of  gravitation,  G,  has  been  found  by 
measuring  the  attraction  between  two  bodies  on  the  surface  of  the 
earth.  When  mr  m^  r,  and  F  are  expressed  in  C.  G.  S.  units,  G  is 
6.6576  x  10~8.  If  we  use  this  value  for  G  and  consider  the  attraction 
between  a  gm  and  the  earth,  that  is,  give  ml  the  value  1,  r  the  value 
of  the  radius  of  the  earth  in  centimetres,  and  F  the  weight  of  a  gm 
in  dynes  at  a  pole  or  978,  the  value  of  w2  deduced  from  the  formula 
for  the  law  of  gravitation  will  be  the  mass  of  the  earth.  This  divided 
by  the  known  volume  of  the  earth  gives  the  mean  density  of  the  earth, 
which  is  thus  found  to  be  5.527.  (Poynting  and  Thomson's  "  Proper- 
ties of  Matter,"  Chapters  II  and  III.) 

57.  Force  in  Simple  Harmonic  Motion.  —  Since  the  accel- 
eration of  a  body  having  a  S.  H.  M.  is 


a  =  -    -=-  }  •  x, 
the  force  acting  on  the  body  when  the  displacement  is  x  is 


Hence  if  a  body  performs  a  vibrating  motion  under  the 
action  of  a  force  which  is  proportional  to  and  in  the  oppo- 
site direction  to  the  displacement,  the  motion  is  S.  H.  M., 
and  if  the  force  at  a  certain  displacement  is  known,  the 
period  T  can  be  calculated.  When  an  elastic  body  such 
as  a  spiral  spring  or  a  bar  is  distorted  in  any  way,  that  is, 
stretched,  bent,  or  twisted,  etc.,  the  force  with  which  it 
resists  the  distortion  and  tends  to  recover  its  form  is  pro- 


FORCE  69 

portional  to  the  distortion  (provided  the  distortion  is  not 
so  great  as  to  cause  a  permanent  change).  This  is  an 
experimental  fact  known  as  Hooke's  law  of  elasticity 
(§127).  Hence  such  a  body  when  distorted  and  set  free 
to  vibrate  performs  S.H.  vibrations,  the  period  depending 
on  the  force  resisting  distortion  and  the  mass  set  into 
vibration.  The  vibrations  of  a  spiral  spring  carrying  a 
weight  and  of  a  tuning-fork  are  examples  that  have  been 
employed  already. 

Exercise  X.    Force  in  S.H.M. 

A  mass  m  is  suspended  by  a  vertical  spiral  spring  and  vibrates  in  a 
vertical  line.  The  motion  is  S.  H.  M.  if  the  resultant  force  acting  on 
the  body  at  any  displacement  x  from  its  position  of  rest  is 
proportional  to  x.  Let  the  length  of  the  spring  when  the 
mass  is  at  rest  be  1.  The  force  Fl  required  to  stretch  the 
spring  to  the  length  I  is  the  weight  of  m.  When  m  is  dis- 
placed through  a  distance  x  (positive  downward)  the  result- 
ant (upward)  force  acting  on  m  is  F2  —  Fv  if  F2  be  the  force 
required  to  stretch  the  spring  to  the  length  I  4-  x.  Hence 
the  motion  will  be  S.  H.  M.  if  (F2  —  F^)  cc  x,  i.e.  if  — - — 
=  a  constant.  If  various  values  are  given  to  F2  and  the  cor- 
responding values  of  x  noted,  a  curve  connecting  F2  (as  or- 
dinate)  and  x  (as  abscissa)  may  be  drawn  and,  if  (F2  —  F}) 
oc  x,  the  curve  will  be  a  straight  line.  From  this  line  a  more 
accurate  value  of  the  constant  ratio  of  (F2  —  FJ  to  x  can  be 
found  and  used  to-calculate  the  period  of  vibration  of  m.  FIQ.  29. 

The  curve  thus  obtained  expresses  the  relation  between 
the  length  of  the  spring  and  the  force  applied  to  it,  and  is  called  the 
"calibration  curve"  of  the  spring.  When  a  spring  has  been  cali- 
brated, it  may,  along  with  its  calibration  curve,  be  used  as  a  spring 
balance  to  weigh  bodies  or  to  apply  known  forces  to  bodies,  and  as 
such  we  shall  have  frequent  occasion  to  use  it. 

The  calibration  of  each  spring  consists  in  determining  the  length 


70  DYNAMICS 

for  each  of  half  a  dozen  or  more  different  weights  attached  and  then 
plotting  a  curve  with  stretching  forces  as  ordinates  and  lengths  as 
abscissae.  The  calibration  of  a  spring  will  be  found  necessary  in 
several  other  exercises.  A  device  that  facilitates  the  calibration  is 
a  vertical  scale  etched  on  mirror  glass  or  on  nickel-plated  steel.  If 
the  spring  be  hung  in  front  of  the  glass  scale,  its  length  between  the 
hooked  ends  can  be  read  by  reflection  without  danger  of  parallax. 
A  numbered  tag  should  be  attached  to  each  spring  calibrated,  and  the 
number  should  be  marked  on  the  calibration  curve. 

The  periods  of  vibration  of  two  masses  attached  to  spiral  springs 
are  to  be  calculated  by  the  above  method  and  then  determined  ex- 
perimentally by  counting  the  number  of  vibrations  in  several  minutes. 

DISCUSSION 

(a)  Limit  to  the  amplitude  if  the  motion  is  to  remain  S.  H.  M. 

(&)  Should  the  masses  of  the  springs  be  taken  account  of  in  the 
calculation  ? 

(c)  How  could  the  vibrations  of  a  spring,  carrying  a  weight,  be 
used  to  find  the  value  of  g  ? 

58.  Composition  and  Resolution  of  Forces  acting  on  a  Par- 
ticle. —  Every  force  has  a  definite  direction  and  a  definite 
magnitude.  Hence  any  number  of  forces  acting  on  a 
particle  can  be  represented  by  lines  in  the  directions  of 
the  forces  and  proportional  in  lengths  to  the  magnitudes 
of  the  forces. 

Since  the  forces  give  rise  to  accelerations  which  are  in  the 
directions  of  the  forces  and  proportional  in  magnitude  to 
the  forces,  a  set  of  lines  that  represent  any  number  of  forces 
applied  to  a  particle  may  be  also  taken  to  represent  the  ac- 
celerations to  which  the  forces  give  rise.  These  accelera- 
tions can  be  compounded  and  resolved  by  methods  already 
stated  (§  9).  Hence  forces  can  be  similarly  compounded 
and  resolved  by  means  of  the  lines  that  represent  them. 


FORCE 


71 


Exercise  XI.    The  Composition  of  Forces 

Composition  of  Two  Forces.  —  Two  spiral  springs  are  attached  to  a 
small  ring.     The  other  ends  of  the  springs  are  tied  to  cords  which 


FIG.  30. 


are  fastened  by  thumb-tacks  to  a  vertical  cross-section  board.     A 
weight  is  hung  from  the  ring  by  means  of  a  cord.     The  board  may 


72  DYNAMICS 

be  levelled  by  using  the  weight  and  cord  as  a  plumb-line.  When  the 
ring  has  come  to  rest  the  projection  of  its  centre  on  the  board  is 
marked  by  a  pin.  The  length  of  each  spring  is  obtained  from  a 
mirror  scale  placed  between  it  and  the  board. 

A  careful  copy  of  the  arrangement  should  then  be  made  on  cross- 
section  paper.  The  direction  of  the  line  representing  each  spring  is 
fixed  by  the  position  of  the  peg  to  which  it  is  attached  and  the  centre 
of  the  ring.  If  the  springs  have  been  already  calibrated,  the  forces 
they  apply  to  the  ring  can  be  deduced  from  their  measure  lengths. 
If  they  have  not  been  calibrated,  each  must  be  hung  vertically  and 
the  weight  required  to  stretch  it  as  in  the  experiment  determined. 
The  resultant  of  the  forces  applied  to  the  springs  can  then  be  found 
graphically  by  completing  the  parallelogram.  It  should  also  be  cal- 
culated by  the  trigonometrical  formula.  The  resultant  should  be 
approximately  equal  and  opposite  to  the  weight  carried  by  the  cord. 

Composition  of  Three  Forces.  —  An  additional  spring  is  attached  to 
the  ring  and  fastened  to  the  board  by  a  peg.  A  drawing  is  made 
on  cross-section  paper  as  before.  The  resultant  is  then  found  (1)  by 
the  polygon  method,  (2)  by  the  analytical  method,  the  angles  the 
forces  make  with  the  horizontal  being  measured  by  a  protractor. 
The  resultant  should  be  approximately  equal  and  opposite  to  the 
weight  carried  by  the  cord  attached  to  the  ring. 

DISCUSSION 

(a)  Meaning  of  resultant  and  proof  of  formulae  used. 

(b)  What  is  the  sum  of  the  vertical  forces  on  the  pegs  equal  to? 
Of  the  horizontal  forces? 

(c)  Calculation  of  the  angle  between  the  cord  and  each  spring  in 
the  first  part  of  the  exercise. 

(cT)  Each  of  the  three  forces  in  the  first  part  of  the  exercise  is 
proportional  to  the  sine  of  the  angle  between  the  other  two. 

(e)  How  does  the  tension  in  each  spring  vary  with  the  inclination 
of  the  spring  to  the  vertical? 

(/)  What  is  the  minimum  strength  of  a  wire  that  will  sustain  a 
heavy  picture  if  the  angle  between  the  two  parts  of  the  wire  be  90°  ? 

(<7)  On  what  does  the  pull  on  a  kite-string  depend  ? 


FORCE  73 

59.  Condition  of  Equilibrium  of  Forces  acting  on  a  Par- 
ticle.—  Any  number  of  forces  acting  on  a  particle  are  said 
to  be  in  equilibrium  when  their  resultant  is  zero. 

Two  forces  are  in  equilibrium  when  they  are  equal  and 
opposite,  and  they  cannot  be  in  equilibrium  unless  they 
are  equal  and  opposite. 

Three  forces  are  in  equilibrium  if  the  resultant  of  two 
of  them  is  equal  and  opposite  to  the  third.  If  the  three 
forces  can  be  represented  by  the  three 
sides  AB,  BC,  CA  of  a  triangle,  the 
sides  being  taken  in  continuous  order, 
then  the  resultant  of  two  of  the  forces 
AB  and  BC  is  equal  and  opposite  to 
the  third  CA,  and  the  forces  are  there- 
fore in  equilibrium. 

Conversely,  if  three  forces  are  in  equilibrium,  and  if 
any  triangle  be  drawn  whose  sides  taken  in  order  are  in 
the  directions  of  the  forces,  then  the  forces  are  propor- 
tional to  the  sides  of  this  triangle.  For  if  any  two  lines 
AB,  BC  be  drawn  to  represent  two  of  the  forces,  the 
resultant  of  these  two  is  represented  by  AC.  Hence  for 
equilibrium  the  third  must  be  represented  by  the  third 
side  CA  of  the  triangle  ABC.  Any  other  triangle  whose 
sides  are  parallel  respectively  to  the  sides  of  the  triangle 
ABC,  that  is,  in  the  directions  respectively  of  the  forces, 
is  similar  to  ABC.  Hence  the  forces  are  proportional 
also  to  the  sides  of  this  second  triangle. 

It  can  be  shown  in  the  same  way  that  a  necessary  and 
sufficient  condition  for  the  equilibrium  of  any  number  of 
forces  is  that  they  should  be  representable  by  the  sides  of 
a  closed  polygon  taken  in  order. 


74  DYNAMICS 

Another  convenient  way  of  stating  the  condition  for 
the  equilibrium  of  any  number  of  forces  is  supplied  by 
the  analytical  method  of  composition  (§  15).  If  the  sums 
of  the  components  of  the  forces  in  three  directions  at  right 
angles  are  Jf,  Y,  Z,  and  if  R  is  the  resultant, 

R*  =  x2  +  r2  +  z2. 

Hence  R  is  0  if  X,  Y,  and  Z  are  each  0. 

Conversely,  if  R  is  0,  Jf,  Y,  and  Z  must  each  be  0,  since 
their  squares  cannot  be  negative. 

Exercise  XII.    The  Triangle  of  Forces 

An  interesting  illustration  of  the  triangle  of  forces  is  its  applica- 
tion to  the  calculation  of  the  forces  that  act  on  the  parts  of  a  jointed 
framework.  As  a  simple  example,  we  may  consider  a  skeleton  frame- 
work ABC  made  up  of  three  spiral  springs  suspended  from  the  verti- 
cal cross-section  board  by  two  other  springs  which  are  attached  to  the 
corners  B  and  C  of  the  triangle.  A  weight  W  attached  to  the  corner 
of  A  will  put  all  the  springs  in  a  state  of  tension.  Let  the  tension  in 
the  sides  of  the  triangle  and  the  forces  applied  to  it  be  Tv  T2,  Tz,  Fv 
Fy  Fs,  as  indicated  in  the  diagram. 

At  the  point  A  three  forces  Tv  Fv  and  T2  act  so  as  to  keep  the 
point  in  equilibrium,  and  they  may  therefore  be  represented  by  any 
triangle  oab  whose  sides  are,  taken  in  order,  in  the  direction  of  the 
forces.  The  forces  T2,  F2,  T3,  that  keep  the  point  B  in  equilibrium, 
may  similarly  be  represented  by  the  sides  obc  of  a  second  triangle 
which  has  one  side  ob  in  common  with  the  first  triangle.  Finally,  by 
joining  c  and  a  we  have  a  triangle  oca  whose  sides  represent  the  forces 
TB,  F3,  Tv  which  act  at  C.  The  figure  oabc  is  sometimes  called  the 
"force-diagram"  of  the  framework  ABC.  From  it  the  magnitudes 
of  all  the  other  forces  can  be  deduced  by  proportion  if  that  of  one  of 
the  forces  be  known. 

Having  carefully  constructed  the  force-diagram  on  cross-section 
paper,  assume  F3  as  known  from  the  magnitude  of  the  suspended 
weight,  and  then  deduce  the  magnitudes  of  the  other  forces  and  tabu- 


FORCE 


75 


late  the  results  and  the  numbers  of  the  springs.  As  a  check  on  the 
results,  carefully  measure  the  lengths  of  the  springs  and  tabulate  the 
results,  and  then  deduce  the  tension  of  the  calibrated  springs  from 


FIG.  32. 

their  calibration  curves.  Calibrate  the  remaining  springs,  or  at  least 
find  the  forces  necessary  to  stretch  them  as  in  the  experiment.  The 
closeness  of  agreement  of  the  results  found  graphically,  and  those 
deduced  from  calibration  of  the  springs,  will  depend  chiefly  on  the 
care  with  which  the  force-diagram  was  drawn. 


76  DYNAMICS 

DISCUSSION 

(a)  Show  that  the  external  forces  applied  to  the  framework  would 
be  in  equilibrium  if  applied  to  a  particle. 

(6)    Show  the  same  for  the  internal  forces  in  the  framework. 

(c)  If  the  sides  of  the  framework  consisted  of  uniform  rods  of 
considerable  weight,  in  what  way  would  the  force-diagram  have  to  be 
modified? 

(rf)  A  cord  fastened  at  the  ends  to  a  support  carries  weights  at 
various  points.  Draw  the  force-diagram  of  the  arrangement  (called 
a  funicular  polygon) . 

(e)  Construction  and  calibration  of  a  simple  form  of  light  balance 
for  letters,  etc.,  on  the  principle  suggested  by  d. 

(/)  Three  forces  acting  on  a  particle  are  represented  by  the  sides 
AB,  A  C,  EC  of  a  triangle.  Find  the  resultant. 

60.  Newton's  Third  Law  of  Motion.  —  "To  every  action 
there  is  an  equal  and  opposite  reaction  or  the  mutual 
actions  of  bodies  are  equal  and  opposite." 

The  truth  of  this  law  is  readily  recognized  in  cases  in 
which  the  bodies  are  at  rest ;  for  instance,  when  one  hand 
is  pressed  against  the  other,  when  a  hand  is  pressed 
against  a  wall,  when  a  hand  supports  a  weight,  when 
two  equal  masses  hang  by  a  cord  that  passes  over  a 
pulley,  when  a  horse  exerts  force  on  a  rope  attached  to 
a  canal  boat  which  is  prevented  from  moving. 

When  applied  to  bodies  in  motion  the  meaning  of  the 
law  is  not  so  obvious.  Consider  the  case  of  the  horse 
and  the  canal  boat  when  both  are  in  motion  with  con- 
stant velocity.  If  the  boat  did  not  pull  backward  on  one 
end  of  the  rope  with  a  force  of  the  same  magnitude  as 
that  with  which  the  horse  pulls  forward  on  the  other  end, 
there  would  be  a  resultant  force  on  the  rope  and  it  would 
move  with  an  acceleration. 


FORCE  77 

Next  suppose  the  horse  and  boat  are  moving  with  an 
acceleration.  If  the  rope  is  so  light  that  its  mass  may 
be  neglected,  then,  if  there  were  an  appreciable  difference 
in  the  magnitudes  of  the  forces  at  its  ends,  it  would  move 
with  a  very  great  acceleration.  If  the  mass  of  the  rope 
is  not  negligible,  we  cannot  any  longer  regard  the  horse 
and  the  boat  as  bodies  acting  and  reacting  directly  on 
one  another,  for  now  there  is  a  body  of  definite  mass 
between  them.  Consider,  however,  a  part  of  the  rope  so 
short  that  its  mass  may  be  regarded  as  negligible;  the 
pulls  at  its  ends  must  be  equal,  for  otherwise  it  would 
move  with  a  very  great  acceleration. 

From  what  has  been  stated  it  will  be  seen  that  (1)  the 
action  and  reaction  spoken  of  are  not  two  forces  acting 
on  the  same  body,  but  the  action  is  a  force  applied  to  one 
body,  the  reaction,  a  force  applied  to  the  other  body; 
(2)  the  bodies  referred  to  are  bodies  directly  in  contact, 
although,  when  there  is  no  relative  acceleration,  or  when 
there  is  an  acceleration  but  the  mass  of  the  intervening 
connection  is  negligible,  bodies  not  directly  in  contact 
may  be  treated  as  if  they  were  in  contact. 

The  reader  should  consider  all  the  actions  and  reactions 
in  the  case  of  the  horse  and  the  canal  boat  (1)  between 
horse  and  ground,  (2)  between  horse  and  rope,  (3)  be- 
tween rope  and  boat,  (4)  between  water  and  boat, 
(5)  between  water  and  ground. 

61.  Stress.  —  Forces  always  occur  in  pairs,  an  action 
and  a  reaction.  The  action  and  reaction  considered  to- 
gether are  called  a  stress.  A  force  is  only  a  partial  aspect 
of  a  stress,  that  is,  a  stress  considered  only  as  regards  its 


78  DYNAMICS 

action  on  a  single  body.  The  complementary  aspect  of 
the  stress  is  the  reaction  on  the  other  body.  In  terms 
of  stress  Newton's  Third  Law  may  be  stated  thus:  "All 
force  is  of  the  nature  of  stress ;  stress  exists  only  between 
two  portions  of  matter,  and  its  effects  on  these  portions 
are  equal  and  opposite." 

There  is  reason  to  believe  that  when  two  bodies  seem 
to  influence  one  another's  motion  without  any  visible  con- 
nection existing  between  them,  e.g.  two  magnets  or  two 
bodies  charged  with  electricity,  the  effect  is  really  due  to 
a  stress  in  an  intervening  medium.  In  the  case  of  mag- 
netized and  electrified  bodies,  the  medium  is  the  ether 
and  the  nature  of  the  stresses  are  to  some  extent  under- 
stood. In  one  important  case,  namely,  the  gravitational 
attraction  between  bodies,  the  nature  of  the  stress  has  not 
yet  been  discovered,  but  the  ether  is  probably  the  medium. 

62.  Transference  and  Conservation  of  Momentum.  —  The 
force  that  a  body  A  exerts  on  a  body  B  is  equal  and 
opposite  to  the  force  that  B  exerts  on  A.  Hence  the 
change  of  momentum  that  A  produces  in  B  in  any  time 
is  equal  and  opposite  to  the  change  of  momentum  that 
B  produces  in  A  in  the  same  time.  If,  therefore,  we 
reckon  momentum  in  one  direction  as  positive,  and  in 
the  opposite  direction  as  negative,  the  mutual  action 
between  two  bodies  produces  no  resultant  change  of 
momentum;  one  suffers  a  decrease  of  positive  momen- 
tum, the  other  an  increase.  Hence  momentum  may  be 
transferred  from  one  body  to  another,  but  the  total  mo- 
mentum is  unchanged  by  the  mutual  action.  This  prin- 
ciple is  sometimes  called  the  conservation  of  momentum. 


FORCE  79 

63.  Changes  of  Velocity  due  to  Mutual  Action  between 
Bodies. — When  two  bodies  act  on  one  another  the  changes 
of  momentum  produced  are  equal  in  magnitude;  hence 
the  changes  of  velocity  are  inversely  as  the  masses.  If  the 
bodies  undergo  equal  changes  of  velocity,  their  masses  are 
equal.  If  the  changes  of  velocity  are  unequal,  the  masses 
are  inversely  as  the  changes  of  velocity.  These  statements 
might  be  taken  as  definitions  of  equality  of  masses  and 
the  ratio  of  two  masses.  They  are  in  reality  the  same 
as  the  definitions  supplied  by  Newton's  Second  Law 
(§  52).  This  way  of  defining  the  ratio  of  two  masses 
leads  at  once  to  an  experimental  method  of  ascertaining 
the  ratio  of  the  masses;  and  while  it  is  not  an  accurate 
practical  method,  an  attempt  to  carry  it  out  will  help  to 
make  the  meaning  of  mass  more  definite. 

Exercise  XIII.    Transference  and  Conservation  of  Momentum 

Two  light  wooden  trays  or  carriers  are  suspended  by  threads  so  as 
to  be  free  to  vibrate  while  remaining  always  horizontal.  If  drawn 
aside  and  released  simultaneously,  they  collide  perpendicularly  at  their 
lowest  positions  and  two  needle-points  attached  to  one  of  them  stick 
into  the  other  and  so  prevent  separation.  Small  scales  are  mounted 
on  a  bar  below  the  carriers  and  pointers  attached  to  the  carriers  move 
along  the  scales  as  the  carriers  swing. 

The  threads  should  be  carefully  adjusted  so  that  each  carrier  moves 
parallel  to  the  line  of  the  scales  and  so  that  the  carriers  just  come 
into  contact  when  hanging  at  rest.  A  convenient  method  of  adjust- 
ment is  to  provide  the  ends  of  each  thread  with  small  rings.  One 
ring  is  attached  to  a  hook  on  the  carrier,  the  other  is  fastened  to  the 
upper  surface  of  the  top-board  by  a  thumb-tack.  After  all  the  threads 
have  been  carefully  adjusted  side-strips  on  the  top-board  are  screwed 
down  so  as  to  prevent  the  threads  getting  out  of  adjustment. 

The  carriers  can  be  released  simultaneously  by  means  of  a  thread 


80 


DYNAMICS 


that  passes  through  four  screw-eyes  (see  Fig.  33)  and  is  attached  to 
both  carriers.     The  carriers  having  been  drawn  aside  to  any  desired 


FIG.  33. 


positions  by  means  of  the  thread,  the  latter  is  attached  to  the  table 
by  a  pin;  when  the  pin  is  pulled  out  the  carriers  are  released. 

The  reader  will  have  no  difficulty  in  showing  that  if  R  be  the 
radius  of  the  circles  in  which  the  carriers  swing  and  x  the  horizontal 


FORCE  81 

distance  of  the  starting  point  of  a  carrier  from  the  lowest  part  of  the 
arc,  its  velocity  at  the  lowest  point  is  v  =  ar\/-^-     Hence  -\/-2  may  be 

calculated  once  for  all;  then  v  can  be  found  for  any  observed  value  of  x. 

Measurement  of  Mass.  First  Method.  —  The  body  of  unknown  mass 
(a  cylinder  of  wood)  is  placed  in  one  carrier  and  known  masses 
("  weights  "  from  a  box  of  weights)  are  placed  in  the  other  carrier 
until  the  carriers,  falling  from  the  same  height,  come  to  rest  on 
colliding.  This  should  be  repeated  several  times,  the  height  of  fall 
being  varied  and  the  known  and  unknown  masses  interchanged.  The 
accuracy  of  the  result  should  be  tested  by  placing  the  unknown  and 
known  masses  on  the  pans  of  an  ordinary  balance. 

Measurement  of  the  Mass  of  a  Carrier.  —  Let  a  known  mass  be  placed 
in  one  of  the  carriers  and  let  the  carriers  be  released  from  such  heights 
that  they  come  to  rest  on  colliding.  Then  let  the  known  mass  be 
placed  on  the  other  carrier  and  the  experiment  repeated  to  ascertain 
whether  the  masses  of  the  carriers  are  equal.  Make  several  careful 
determinations  of  the  mass  of  each  carrier. 

Measurement  of  Mass.  Second  Method.  —  Suppose  only  a  single 
known  mass  is  available.  Place  the  body  of  unknown  mass  in  one 
carrier  and  the  known  mass  in  the  other  and  find  the  heights  from 
which  the  carriers  must  be  released  so  that  they  shall  come  to  rest 
on  impact.  Interchange  and  repeat. 

Measurement  of  Mass.  Third  Method.  —  Place  the  unknown  mass 
in  one  carrier  and  a  known  mass  in  the  other.  Allow  the  carriers 
to  fall  from  equal  heights  and  find  the  velocity  of  the  combined 
mass  after  collision.  Next  let  one  carrier  impinge  on  the  other  at 
rest.  Other  combinations  of  initial  velocities  may  be  tried.  From 
each  the  unknown  mass  is  calculated. 

DISCUSSION 

(a)  Sources  of  error. 

(b)  Meaning  of  equality  of  mass. 

(c)  Meaning  of  ratio  of  two  masses. 

(d)  Proof  of  formula  used  in  calculating  v. 

(e}  Is  the  present  method  of  comparing  masses  independent  of  the 
assumption  that  weight  is  proportional  to  mass? 


82  DYNAMICS 

(/)  On  what  ground  is  it  assumed  that  the  carriers,  if  released  at 
the  same  time,  always  meet  at  the  lowest  points  of  their  swings  ? 

(#)  Measurement  of  the  velocity  of  a  bullet  by  attaching  the  gun 
to  a  heavy  pendulum  and  noting  the  deflection  of  the  pendulum  when 
the  gun  is  fired. 

64.  The  Force  required  to  make  a  Body  revolve  in  a  Circle. 
—  We  have  already  seen  (§  33)  that  when  a  body  revolves 
in  a  circle  of  radius  r  with  linear  speed  s  it  has  an  accel- 

s2 
eration  —  toward  the  centre.     To  give  it  this  acceleration, 

a  force  directed  toward  the  centre  must  be  applied  to  it. 

s2 
Since  F=  ma,  and  a  =— ,  the  necessary  force  toward  the 

centre  is 

F=—- 
r 

To  this  force  there  is  an  equal  and  opposite  reaction 
due  to  the  inertia  of  the  body.  This  reaction  of  a  revolv- 
ing body  against  acceleration  toward  the  centre  is  called 
"  centrifugal  force."  It  must  not  be  thought  of  as  a  force 
acting  on  the  body ;  the  only  force  acting  on  the  body  is 
toward  the  centre,  the  "  centrifugal  force  "  is  the  reaction 
of  the  body  in  a  direction  away  from  the  centre.  Thus 
when  a  stone  is  whirled  around  at  the  end  of  a  string,  the 
force  applied  to  the  body  is  a  pull  toward  the  centre  pro- 
duced by  the  hand ;  the  "  centrifugal  force "  is  the  out- 
ward pull  the  body  exerts  on  the  hand. 

Exercise  XIV.    Acceleration  and  Force  in  Uniform  Circular  Motion 

Apparatus.  —  A  horizontal  spiral  spring  connects  a  body  that  ro- 
tates in  a  horizontal  circle  to  a  vertical  steel  axis  supported  on  needle- 
points. The  weight  of  the  rotating  body,  a  lead  block,  is  borne  by  a 
cord  attached  to  a  horizontal  rod  that  passes  through  the  steel  axis. 


FORCE 


83 


The  cord  is  given  the  form  of  a  V  in  order  to  keep  the  body  from 
swaying  backward  and  forward  in  the  arc  of  the  circle  described  by 
the  body.  When  the  vertical  axis  rotates  steadily,  the  spring  is 
stretched  and  the  lead  block  moves  in  a  circle.  If  the  plane  of  the 


FIG.  34. 


cord  be  vertical,  the  horizontal  central  force  acting  on  the  lead  block 
will  equal  the  tension  of  the  spring. 

The  rotation  of  the  axis  is  produced  by  a  silk  thread  that  is 
wrapped  around  the  axis ;  the  thread  passes  over  a  pulley  and  carries 
a  weight.  The  thread  is  attached  to  a  collar  that  is  adjustable  along 


84  DYNAMICS 

the  axis ;  a  small  ring  attached  to  the  thread  hangs  on  a  peg  attached 
to  the  collar,  so  that  when  the  thread  is  wholly  unwrapped  it  becomes 
detached  and  then  the  axis  rotates  at  a  constant  rate  (except  for  the 
small  effect  of  friction). 

For  recording  the  speed  of  rotation  at  any  time  the  axis  carries  a 
horizontal  circular  disk  around  which  a  strip  of  paper  is  fastened.  A 
fine-pointed  camel's-hair  brush  attached  to  the  knife-edge  of  a  short 
adjustable  pendulum  sweeps  across  the  strip  of  paper  as  the  pendulum 
vibrates ;  when  inked  the  brush  records  the  vibrations  of  the  pendu- 
lum on  the  strip  of  paper. 

Adjustments.  —  After  the  weight  has  been  hung  in  position  and 
before  the  spring  is  attached,  a  pin  is  fixed  vertically  in  the  bottom 
of  the  lead  block.  Another  pin  is  fixed  in  the  top  of  a  wooden  block 
that  rests  on  the  table  just  below  the  lead  block.  The  wooden  block 
is  placed  on  the  table  so  that  the  pins  are  in  the  same  vertical  line. 
The  spring  is  then  attached.  It  should  be  horizontal  when  so 
stretched  that  the  pins  are  opposite  one  another.  This  adjustment 
can  be  made  with  sufficient  accuracy  by  holding  the  spring  stretched 
and  testing  it  by  a  small  level  held  above  it.  The  length  of  the  pen- 
dulum should  be  carefully  adjusted  so  that  the  pendulum  beats  sec- 
onds as  tested  by  counting  the  vibrations  in  two  or  three  minutes. 
Until  the  pendulum  is  to  be  used  for  obtaining  a  record,  it  is  held  out 
of  the  vertical  by  a  small  lever.  The  silk  thread  should  be  of  such  a 
length  that  it  becomes  detached  just  before  the  pendulum  reaches  the 
floor.  As  the  weight  descends  and  the  speed  of  the  axis  increases, 
the  spring  lengthens  until  the  pins  come  opposite  one  another ;  this  is 
the  moment  at  which  the  thread  should  become  detached.  After  a 
few  trials  the  proper  height  from  which  to  release  the  weight  (as  in- 
dicated by  a  vertical  metre-stick)  is  readily  ascertained. 

Measurements.  —  When  the  preceding  adjustments  have  been  com- 
pleted, the  weight  is  allowed  to  descend.  The  brush  is  then  inked, 
and  as  soon  as  the  thread  becomes  detached  the  pendulum  is  released. 
After  one  vibration  the  pendulum  is  arrested  so  that  the  record  may 
not  be  confused.  The  order  in  which  the  two  lines  on  the  paper 
were  made  can  be  ascertained  from  their  slope.  This  record  should 
be  numbered  in  lead  pencil  and  then  several  subsequent  records  may 


FORCE  85 

be  obtained  in  the  same  way  on  the  same  strip  of  paper  without  any 
confusion.  When  a  sufficient  number  to  give  a  good  average  have 
been  obtained,  the  brush  should  be  allowed  to  inscribe  a  complete 
circle  on  the  paper,  the  pendulum  remaining  at  rest.  The  paper  may 
then  be  removed.  A  little  thought  will  show  how  the  speed  of  rota- 
tion may  be  deduced  from  the  records. 

If  the  spring  has  been  calibrated,  its  tension  can  be  deduced  from 
its  length  and  the  calibration  curve.  If  not  yet  calibrated,  its  length 
when  stretched  is  carefully  measured  and  it  is  then  removed  and  cali- 
brated. Or  the  following  procedure  may  be  adopted :  attach  a  cord 
to  the  block  so  as  to  stretch  the  spring  and  allow  the  cord  to  hang 
over  a  pulley  clamped  to  the  framework,  so  that  the  part  of  the  cord 
between  the  lead  block  and  the  pulley  is  horizontal  and  then  place 
such  weights  in  a  pan  carried  by  the  cord  that  the  pins  come  into 
line.  While  the  apparatus  is  in  this,  position,  the  radius  of  the  path 
described  by  the  centre  of  the  block  may  be  obtained  by  means  of  the 
beam-compass,  measurements  being  made  of  both  the  inside  and  the 
outside  distances  of  axis  and  block  and  the  mean  taken. 

From  the  speed  of  rotation  (in  radians  per  second)  and  the  radius 
of  the  circle  described  by  the  centre  of  the  block,  the  acceleration 
toward  the  centre  is  calculated.  From  this  and  the  mass  of  the 
block  the  central  force  is  deduced  and  compared  with  the  tension  of 
the  spring. 

The  same  process  should  be  repeated  with  a  spring  of  different 
stiffness,  again  with  a  block  of  different  mass,  and  again  with  a 
different  radius  of  rotation  (obtained  by  slipping  the  horizontal  arm 
through  the  vertical  axis). 

Second  Method.  —  An  interesting  variation  of  the  above  that  gives 
good  results  and  dispenses  with  recording  disk,  pendulum,  and  thread 
and  weight  may  be  briefly  sketched.  The  axis  is  set  into  rotation  by 
slight  impulses  from  the  thumb  and  forefinger  on  the  lower  end  of 
the  axis.  By  the  same  means  it  is  kept  in  rotation  at  such  a  rate 
that  when  the  moving  pin  passes  the  stationary  one  they  are  as  nearly 
as  possible  in  line.  Only  very  slight  impulses  are  needed,  as  the  fric- 
tion of  the  bearings  is  very  slight.  When  the  right  speed  of  rotation 
has  been  obtained  and  can  be  kept  up,  the  speed  of  rotation  can  be 


86  DYNAMICS 

found  by  counting  the  number  of  revolutions  in  a  given  time,  say 
two  or  three  minutes.  If  a  stop-watch  is  used,  this  will  present  no 
difficulty.  If  a  clock  or  chronometer  circuit  *  is  used  for  time,  each 
minute  may  be  regarded  as  beginning  and  ending  at  the  first  tick 
after  a  silence.  Begin  counting  passages  after  the  tick  that  indicates 
the  beginning  of  a  minute,  call  this  the  zero  passage,  and  continue 
until  the  end  of  the  two  or  three  minutes.  To  obtain  a  good  mean, 
this  should  be  repeated  several  times. 

DISCUSSION  AND  PROBLEMS 

(a)  Meaning  and  deduction  of  formula. 
(5)  Meaning  of  "  centrifugal  force." 

(c)  Force  acting  on  vertical  axis. 

(d)  Effect  of  spring  not  being  horizontal. 

(e)  Effect  of  supporting  cord  not  being  vertical. 

(/)  How  the  mass  of  the  rotating  body  could  be  deduced  from  this 
experiment. 

(g)  Direction  and  magnitude  of  whole  resultant  force  on  rotating 
body. 

(h)  At  what  inclination  to  the  vertical  would  the  supporting  cord 
stand  if  the  body  rotated  at  the  same  speed  but  the  spring  were  absent? 

(i)  What  angular  velocity  must  a  boy  give  to  a  sling  of  80  cm. 
length  in  order  that  the  stone  may  not  fall  out  when  it  is  at  the 
highest  point? 

(/)  The  centre  of  the  moon  is  about  60  times  the  earth's  radius 
from  the  centre  of  the  earth,  and  it  revolves  once  in  27  days  8  hours. 
Compare  its  acceleration  with  that  of  a  body  allowed  to  fall  near  the 
surface  of  the  earth.  Test  the  law  of  gravitation. 

(fc)  State  the  formula  for  "centrifugal  force"  in  gravitational 
units. 

65.  The  Conical  Pendulum.  —  A  ball  is  attached  to  the 
end,  P,  of  an  arm,  AP,  that  is  pivoted  at  A  to  a  vertical 

*  A  circuit  containing  a  relay  or  sounder  and  connected  with  a  chro- 
nometer or  clock  in  such  a  way  that  the  relay  sounds  once  per  second, 
but  fails  to  sound  at  the  completion  of  each  minute. 


FOECE 


87 


axis  that  rotates  with  an  angular  velocity  «.  P  revolves 
in  a  circle  of  radius  r,  and  AP  describes  a  cone  of  height 
h ;  hence  P  is  acted  on  by  a  force  muPr  directed  toward  (7, 
the  centre  of  the  circle.  The  tension  in  AP  may  be 
resolved  into  a  vertical  component  that  A 
supports  the  weight,  mg,  of  the  ball,  and 
a  horizontal  component  that  supplies  the 
force  mco2r  in  the  direction  PC.  Hence  h 
from  the  triangle  PC  A  we  get 

r 
h 


mg 


-* 


FIG.  35. 

Watt's  governor  for  a  steam  engine  is  essentially  a 
double  conical  pendulum  applied  to  the  regulation  of  a 
steam  valve. 

Exercise  XV.    The  Conical  Pendulum 

The  same  apparatus  is  used  as  in  the  preceding  exercise,  except 
that  the  cross-arm  is  removed  and  the  short  bar  that  carries  the 
V-cord  is  placed  in  a  hole  in  the  vertical  axis.  The  block  with  its 
vertical  pin  is  adjusted  until  the  end  of  the  pin  in  the  revolving  ball 
just  comes  above  the  fixed  pin,  when  the  weight  reaches  the  floor  and 
the  thread  becomes  detached  from  the  axis.  This  adjustment  having 
been  made,  the  apparatus  is  brought  to  rest  by  a  little  pressure  of 
thumb  and  forefinger  .on  the  vertical  axis,  the  thread  is  rewound  and 
the  weight  again  allowed  to  descend  from  the  same  height,  and  when 
the  weight  reaches  the  floor  the  pendulum  is  released  and  a  record  of 
speed  obtained.  Or  the  second  method  of  procedure  of  the  preceding 
exercise  (dispensing  with  the  recording  disk)  may  be  employed. 

The  value  of  h  cannot  readily  be  found  by  direct  measurement. 
Perhaps  the  best  way  is  to  measure  AP  and  PC  (Fig.  35),  and  de- 
duce A.  These  distances  must  be  measured  from  centre  of  ball  to 


88  DYNAMICS 

centre  of  steel  axis.  Hence  for  each  distance  two  measurements 
must  be  made  with  the  beam-compass,  an  inside  measurement  and 
an  outside  measurement,  the  ball  being  meanwhile  held  by  a  cord 
that  passes  round  it  and  is  attached  to  the  wooden  support.  The 
value  of  to,  calculated  from  h  and  g,  should  agree  closely  with  the 
experimental  value. 

By  changing  the  initial  height  from  which  the  weight  descends, 
different  values  of  <o  may  be  tried. 

DISCUSSION 

(a)  Sources  of  error. 

(&)  What  is  the  length  of  a  simple  pendulum  that  vibrates  once 
during  a  revolution  of  the  conical  pendulum  ? 

(c)  What  motion  does  the  ball  seem  to  have  if  viewed  from  a  great 
distance  in  the  plane  of  revolution  ? 

(d)  The  vertical  distance  of  a  governor -ball  below  the  pivot  varies 
inversely  as  the  square  of  the  velocity  of  revolution. 

(e)  Calculate  the  tension  of  the  suspension. 

(/)  If  several  pendulums  of  different  lengths  were  attached  to  A, 
how  would  they  hang  when  the  rotation  became  steady  ? 

(#)  At  what  angle  does  a  bicyclist  tilt  his  bicycle  in  going  around 
a  curve  ? 

(h)  How  strong  must  the  spokes  of  a  fly-wheel  be  to  be  able  to 
stand  all  the  strain  without  aid  from  the  rim  ? 

(i)  How  strong  must  the  rim  be  to  be  able  to  stand  all  the  strain 
without  aid  from  the  spokes? 

66.  Friction.  —  When  the  surfaces  of  two  solids  are  in 
contact  there  is  a  resistance  to  sliding.  This  resistance  is 
called  friction.  If  a  force  tending  to  produce  sliding  be 
applied  to  one  of  the  bodies,  the  other  being  kept  at  rest, 
sliding  will  not  take  place  unless  the  force  be  above  a  cer- 
tain value.  For  forces  less  than  this  critical  value,  the 
friction  just  equals  the  applied  force  and  no  sliding  takes 
place.  The  force  just  necessary  to  produce  sliding  is  a 
measure  of  the  maximum  static  friction  between  the  sur- 


FORCE  89 

faces.  So  measured  the  maximum  static  friction  is  found 
to  be  proportional  to  the  perpendicular  pressure  between 
the  surfaces,  at  least  throughout  a  considerable  range  of 
pressure.  The  ratio  of  the  maximum  static  friction,  .F,  to 
the  perpendicular  pressure,  P,  between  the  surfaces  is 
called  the  coefficient  of  static  friction,  n,  or 


When  sliding  has  once  begun  it  is  found  that  a  smaller 
force  will  suffice  to  continue  the  motion.  The  force  that 
will  just  maintain  the  motion  is  a  measure  of  the  kinetic 
friction  between  the  surfaces,  and  the  ratio  that  it  bears  to 
the  normal  pressure  is  called  the  coefficient  of  kinetic  friction 
between  the  surfaces.  The  general  results  of  experiments 
on  kinetic  friction  may  be  summarized  as  follows  : 
(1)  the  ratio  of  the  kinetic  friction  to  the  pressure,  i.e. 
the  coefficient  of  kinetic  friction,  is  practically  constant 
through  a  wide  range  of  variation  of  pressure,  (2)  the 
ratio  is  also  practically  independent  of  the  speed  of  sliding 
provided  the  latter  be  not  very  small,  (3)  when  the  speed 
is  very  small  and  decreases  toward  zero,  the  friction 
increases  and  approaches  more  and  more  the  magnitude  of 
the  maximum  static  friction  and  at  indefinitely  small  speeds 
the  two  are  equal. 

The  coefficient  of  kinetic  friction  between  surfaces  of 
wood  depends  on  the  materials,  varying  between  .25  and 
.50.  In  the  case  of  metal  surfaces  it  lies  between  .15 
and  .20. 

67.  Motion  on  an  Inclined  Plane.  —  A  body  on  an  inclined 
plane  is  acted  on  by  two  forces,  gravity  and  friction.  If 


90 


DYNAMICS 


the  mass  of  the  body  is  m  and  the  inclination  of  the  plane 
to  the  horizontal  is  i,  the  weight,  mg,  of  the  body  may  be 
resolved  into  the  component  mg  sin  i  parallel  to  the  plane, 
and  the  component  mg  cos  i  perpendicular  to  the  plane. 
If  i  is  such  that  the  body  just  begins  to  move  when 
released,  mg  sin  i  is  just  equal  to  the  maximum  static 
friction ;  and  since  the  pressure  between  the  body  and  the 

plane  is  mg  cos  i,  the  coefficient  of 

static  friction  is 


n  — 


mg  sin  ^ 


.  =  tan  i. 


mg  cos  i 

From  this  n  may  be  determined. 
If  i  is  such  that  the  body  slides 
downward  with  an  acceleration  a, 
the  whole  force  parallel  to  the 
plane  must  equal  ma.  This  force  consists  of  the  com- 
ponent of  gravity  down  the  plane,  mg  sin  i,  and  the  force 
of  friction  in  the  opposite  direction,  or  n1  -  mg  cos  z,  n' 
being  the  coefficient  of  kinetic  friction.  Hence 
mg  sin  i  —  n'mg  cos  i  =  ma. 

.«.  n'  =  tan  i sec  i. 

9 

Hence  if  i  and  a  be  measured,  n'  can  be  deduced.  The 
value  obtained  for  n1  will,  of  course,  be  its  value  for  the 
particular  speed  at  which  the  body  is  moving  when  the 
acceleration  is  a. 

Exercise  XVI.    Friction 

(1)  The  coefficient  of  static  friction  of  a  pine  block  on  a  pine  board 
is  found  by  adjusting  the  latter  to  such  an  inclination  that  the  block 
just  slides  when  released.  The  same  should  be  done  with  different 


FORCE 


91 


weights  placed  on  the  block  to  show  how  far  the  coefficient  is  inde- 
pendent of  the  pressure.  The  results  of  different  trials  will  be  more 
consistent  the  more  uniform  the  surfaces  are.  Surfaces  that  have 
been  freshly  sandpapered  and  well  brushed  will  give  good  results. 


FIG.  37. 

(2)  To  find  the  coefficient  of  kinetic  friction  of  the  same  surfaces, 
adjust  the  board  to  a  high  inclination  (about  60°  to  the  horizontal), 
and  attach  a  strip  of  glass  coated  with  soap  (bon  ami)  to  the  block 
by  means  of  thumb-tacks.  Place  a  tuning-fork  with  stylus  (as  in 
Exercise  VI)  so  that  the  stylus  will  draw  a  curve  on  the  glass  as  the 
block  descends.  A  second  stylus  attached  to  the  inclined  plane  should 
be  adjusted  so  that,  as  the  block  descends,  it  will  trace  a  (nearly 
straight)  line  that  passes  exactly  through  the  middle  of  the  waves 
traced  by  the  first  stylus.  The  two  styli  should  be  as  close  together 


92  DYNAMICS 

as  practicable,  and  placed  so  that,  when  both  are  at  rest,  they  trace 
but  a  single  line  on  the  glass  as  the  block  descends. 

To  find  the  acceleration  of  the  block,  measure,  along  the  line  traced 
by  the  fixed  stylus,  the  length  of  several  successive  groups  of  four 
waves  each.  The  measurement  may  be  made  by  inverting  the  glass 
on  a  millimetre  scale  (the  tape  of  Exercise  V  will  do),  and  the  read- 
ings should  be  as  exact  as  possible.  The  time  for  each  group  is  known 
from  the  frequency  of  the  fork.  Thus  the  mean  velocity  in  each 
group  is  readily  found,  and  a  value  of  the  acceleration  may  be  deduced 
from  each  two  successive  groups.  As  many  different  values  as  possible 
should  be  obtained  and  averaged,  and  the  coefficient  of  friction  cal- 
culated from  the  average. 

DISCUSSION 

(a)  How  far  did  the  results  show  that  the  coefficient  of  static 
friction  is  independent  of  the  pressure? 

(6)  Did  the  results  indicate  any  variation  of  coefficient  of  friction 
with  velocity  ? 

(c)  Why  did  the  straight  line  need  to  be  drawn  exactly  through 
the  middle  of  the  wave  line  ? 

(e?)  If  the  block  and  board  were  horizontal,  and  the  former  were 
acted  on  by  a  horizontal  force  equal  to  its  weight,  with  what  accelera- 
tion would  it  move  and  how  far  would  it  go  in  10  sec.  ? 

(e)  If  the  force  in  (d)  act  at  the  centre  of  the  block  at  an  angle  a 
with  the  horizontal,  what  will  a  be  if  the  block  just  start,  (1)  when 
the  force  is  a  pull ;  (2)  when  the  force  is  a  push  ? 

(/)  A  body  sliding  down  the  length  of  a  smooth  plane  attains  the 
same  velocity  as  if  it  fell  vertically  the  height  of  the  plane ;  but  this 
is  not  so  if  the  plane  is  rough. 

68.  Dimensions  of  Force  and  Momentum.  —  The  unit  of 
momentum  is  the  momentum  of  a  body  of  unit  mass 
moving  with  unit  velocity.  It  therefore  varies  directly 
as  the  unit  of  mass  and  also  directly  as  the  unit  of  velocity 

or  (room)  oc  C3f  )(F);  but  ^  26^  (TO  ^C^7"1);  hence 
.     Instead  of  the  sign  of  variation,  oc, 


FORCE  93 

we  may  use  the  sign  of  equality,  meaning  equality  of 
dimensions,  and,  as  this  is  the  more  common  method,  we 
shall  hereafter  use  it. 

The  unit  of  force  is  the  force  that  gives  unit  of  mass 
unit  acceleration  ;  it  therefore  varies  directly  as  the  unit 
of  mass,  and  also  directly  as  the  unit  of  acceleration,  or 


It  is  evident  that  these  dimensional  relations  can  be 
derived  directly  from  equations  connecting  the  quantities 
of  unknown  dimensions  and  other  quantities  of  known 
dimensions.  Thus  from  momentum  =  ma  we  get  (mom)  = 
(MLT-V),  and  from  F=  ma  we  get  (^)  =  (MLT~*).  In 
deriving  dimensional  relations  by  this  method  we  neglect 
numerical  constants,  since  they  do  not  depend  on  the 
fundamental  units  or  are  of  zero  dimensions. 

REFERENCES  FOR  CHAPTER  V 

Mach's  "  Science  of  Mechanics." 

Macgregor's  "Kinematics  and  Dynamics,"  Part  II,  Chapters  I 
and  II. 

Lodge's  "Pioneers  of  Science"  (historical). 


CHAPTER  VI 

MOMENT  OF  FORCE 

69.  In  the  preceding  chapter  we  have  considered  the 
motion  of  translation   produced  by  forces   acting   on   a 
particle.     When  a  force  produces  rotation  of  a  body,  the 
magnitude  of  the  effect  depends  on  something  more  than 
the  magnitude  and  direction  of  the  force  and  the  magni- 
tude of  the  mass  moved.     Every  one  knows  that,  to  set  a 
heavy  wheel  in  rotation,  the  force  should  be  applied  as  far 
from  the  axis  of   rotation  as  possible.      The  importance 
of  a  force  as  regards  rotation,  or  the  moment  *  of  the  force 
as  it  is  called,  depends  on  the  magnitude  and  direction  of 
the  force,  and  also  on  its  distance  from  the  axis  of  rotation. 
The  opposition  offered  by  the  inertia  of  the  wheel  is  greater, 
the  farther,  on  the  whole,  the  mass  of  the  wheel  is  from 
the  axis  of  rotation.     In  other  words,  the  importance  of 
inertia  as  regards'  rotation,  or  the  moment  of  inertia  as  it 
is  called,  depends  on  the  distances  of  the  parts  of  the  body 
from  the  axis  of  rotation.     In  the  following  sections  we 
shall  arrive  at  more  precise  definitions  of  moment  of  force 
and  moment  of  inertia. 

70.  Moment  of  Force  and  Moment  of  Inertia.  —  Consider 
a  particle  P,  of  mass  m,  free  only  to  rotate  about   an 

*  The  word  moment,  in  the  sense  of  importance,  occurs  in  such  phrases 
as  "  a  matter  of  no  moment." 

94 


MOMENT  OF  FORCE  95 

axis,  .A,   in   a   circle  whose   centre    is   0  and   radius  r. 

Let  a  force  F  act  on  P  in  the  plane  of  the  circle  (7, 

then  the  only  part  of  F 

that  can  affect  the  motion 

of  P   is  the  component 

tangential  to  the  circle. 

Let  the    direction   of  F 

make  an    angle    9    with 

the  tangent  to  the  circle  ; 

J.-U  xt,  £C        j.'  FlG'   38' 

then  the    effective   com- 

ponent of  F  equals  F  cos  6.     If  the  linear  acceleration 

of  P  be  a, 


From  (7  drop  a  perpendicular,  p,  on  the  line  in  which  F 
acts.     Then  the  angle  between  p  and  the  radius  through 

P 
P  is  also  0  and  cos  6  =  -• 


Since  this  is  a  case  of  rotation  only,  it  is  more  properly 
stated  in  terms  of  the  angular  acceleration  a  of  the  par- 
ticle P.  Now  a  —  ar 


a. 


Hence  the  effectiveness  of  the  force  F  in  producing 
rotation  is  measured  by  Fp.  The  product  of  the  force  F 
by  its  perpendicular  distance  from  the  axis  is  called  the 
moment  of  the  force  about  that  axis. 

If  F  be  not  perpendicular  to  the  axis,  it  may  be  resolved 
into  a  component  parallel  to  the  axis  and  a  component  per- 
pendicular to  the  axis.  The  only  part  of  F  that  will  tend 
to  produce  rotation  about  the  axis  will  be  the  component 
perpendicular  to  the  axis,  and  the  moment  of  F  about  the 


96  DYNAMICS 

axis  will  be  the  component  perpendicular  to  the  axis  mul- 
tiplied by  the  distance  of  this  component  from  the  axis. 

If,  instead  of  being  attached  to  the  axis  A,  the  particle 
m  be  entirely  free,  the  components  of  F  parallel  to  A 
and  along  CP  will  cause  accelerations  in  those  directions, 
but  the  result  as  regards  rotation  about  A  will  not  be 
changed. 

The  multiplier  of  a  in  the  above  equation,  namely  mr2, 
depends  on  the  mass  of  the  particle  and  the  distance  from 
the  axis.  The  product  mr2  is  called  the  moment  of  inertia 
of  the  particle  m  about  the  axis. 

71.  Rotation  of  a  Rigid  Body.  —  As  the  simplest  case  of 
a  rigid  body,  imagine  two  particles  ml  and  m2  in  the  plane 
of  the  paper  connected  together  by  a  rod  whose  mass  may 
be  neglected,  and  suppose  that  they  are  only  free  to  rotate 
about  an  axis  through  0  perpendicular  to  the  plane  of  the 
paper.  Then  at  any  moment  the  particles  must  have  the 
same  angular  velocity  and  angular  acceleration  about  0. 
Let  forces  Fl  and  F%  act  in  the  plane  of  the  paper  on 
the  particles  m1  and  m2  respectively,  and  let  perpendiculars 
from  0  on  Fl  and  _F?  be  p1  and  pz  respectively.  Let  the 

perpendicular  from  C  on  the 
connecting  rod  be  p.  Then 
since  the  stress  T  in  the  rod 
acts  in  opposite  directions  on 
the  particles, 


—  Tp  = 
Hence, 

(w^2  +  W2r22)  a. 


MOMENT  OF  FOECE  97 

Evidently  we  can  extend  the  method  here  used  to  any 
number  of  particles  rigidly  connected  together  and  there- 
fore to  a  rigid  body.     Hence  for  a  rigid  body 
I,Fp  =  a  -  Zmr2. 

Swr2,  or  the  sum  of  the  moments  of  inertia  of  the  parti- 
cles of  the  body  about  a  particular  axis,  is  called  the 
moment  of  inertia  of  the  body  about  that  axis.  It  evi- 
dently depends  only  on  the  mass  and  form  of  the  body. 

^Fp,  or  the  sum  of  the  moments  of  the  various  external 
forces  about  a  particular  axis,  is  the  total  moment  of  force 
about  that  axis.  Denoting  the  moment  of  inertia  of  the 
body  about  a  certain  axis  by  I  and  the  total  moment  of 
force  about  the  axis  by  (7, 

C=Ia. 

If  the  forces  F1  and  F^  do  not  act  in  the  plane  of  rota- 
tion, the  only  parts  of  them  we  need  consider  are  their 
components  in  the  plane  of  rotation,  and  these  are  the 
only  parts  that  contribute  to  the  moment  of  force,  (7,  about 
the  axis  of  rotation.  Hence  the  result  is  unchanged.  If 
m1  and  w2  rotate  about  the  axis  in  two  different  planes 
perpendicular  to  the  axis,  the  only  difference  in  the  above 
proof  will  be  that  the  stress  jPwill  be  inclined  to  the  axis; 
but  since  its  component  in  a  plane  perpendicular  to  the 
axis  will  still  be  a  pair  of  equal  and  opposite  forces  acting 
on  m^  and  w2  respectively,  the  proof  will  still  hold  good. 
Hence  the  formula  0=  Iu<  applies  to  a  body  of  any  shape. 
The  analogy  between  the  formula  for  rotation  and  the 
formula,  F=  Ma,  for  translation  should  be  noted. 

72.  Moment  of  Inertia  of  a  Uniform  Rod.  —  Let  the  length 
of  the  rod  be  L  and  its  mass  M.  We  shall  suppose  that 


98  DYNAMICS 

the  rod  is  of  constant  cross-section  and  that  its  thickness 
is  small  compared  with  its  length.*  Let  the  length,  L,  of 
the  rod  be  divided  up  into  a  large  number,  JV,  of  short 

M 
equal  parts  each  of  mass  —  and 

L 

length  —  .     The  nth  part  reck- 
N 

oning    from    one    end   is   at   a 
distance  ^—  from  that  end,  and  its  moment  of  inertia 

jy 


about  that  end  is  r^  •     Hence,  summing  up  for 

all  values  of  n  from  1  to  JV,  the  total  moment  of  inertia  is 


If  -ZVbe  supposed  indefinitely  large,  —  =  0. 

.-.  I=±  ML*. 

To  deduce  the  moment  of  inertia  of  a  rod  about  its  centre 
we  have  only  to  apply  this  formula  to  the  two  halves  of 


*  The  student  who  is  familiar  with  the  differential  and  integral  calcu- 
lus may  substitute  the  following  : 

Let  p  be  the  mass  of  unit  length  of  the  rod.  The  mass  of  a  short  length 
dr  is  pdr  and  its  moment  of  inertia  about  one  end  of  the  rod  is  pr2dr. 

=  i  ML*. 


MOMENT  OF  FOECE 


99 


73.   Moment  of  Inertia  of  a  Uniform  Rectangular  Disk.  — 

(1)  About  an  axis  in  the  plane  of  the  disk  and  bisecting  the 
sides  whose  length  is  a.     Suppose  the  whole  rectangle  di- 
vided  up   into   narrow   strips    parallel    to    the    sides   a. 
Applying  to  these  the  for- 
mula for  the  moment    of 
inertia  of  a  rod  and  adding 
for  all  the  strips,  we  get 


(2)  About  an  axis  in  the 
plane  of  the  disk  and  bi- 
secting the  sides  whose  length 


(3)  About  an  axis  through  the  centre  perpendicular  to  the 
disk.  The  moment  of  inertia  of  a  particle,  m,  whose  dis- 
tance from  the  centre  is  r  is  mr2.  If  r^  and  r2  be  the  dis- 
tances of  m  from  the  axes  considered  in  (1)  and  (2), 


Hence  if  we  sum  up  both  sides  of  the  equation  for  all  par- 
ticles in  the  disk,  we  get 


NOTE.  —  It  is  obvious  that  the  method  here  used  for  finding  the 
moment  of  inertia  about  an  axis  perpendicular  to  the  plane  of  the 
disk  applies  to  a  disk  of  any  form.  If  its  moment  of  inertia  about  any 
two  rectangular  axes  in  the  plane  of  the  disk  be  II  and  /2,  then  its  mo- 
ment of  inertia  about  a  third  axis  passing  through  the  intersection  of 
the  first  two  and  perpendicular  to  the  planes  of  the  disk  is  /  =  II  +  72. 


100 


DYNAMICS 


74.   Moment  of  Inertia  of  a  Rectangular  Block.  —  The 

block  may  be  divided  up  into  disks  parallel  to  one  face. 
The  moment  of  inertia  of  each  disk  about  an  axis,  through 
its  centre  and  perpendicular  to  its  plane,  is  given  by  (3) 
of  the  preceding  section.  Hence,  adding  for  all  the  disks 


M  being  the  mass  of  the  block,  and  a  and  b  the  sides  of 
the  face  to  which  the  axis  is  perpendicular. 

75.  Moment  of  Inertia  of  a  Uniform  Circular  Disk.  —  Let 
the  radius  of  the  disk  be  R,  and  its  mass  M.*  Suppose 

the  disk  divided  up  into  a 
large  number,  N,  of  con- 
centric rings,  each  of  width 

R 

—     The  area  of  the  nth 

N 

ring    from    the    centre   is 

7?        7? 

f\  J\i          J-li     cy      ll/  7">2 

Now  TrR2  is  the  area  of 
the  whole  disk.  Hence 

the  area  of  the  ring  is  the 

9  vt 
FIG.  42.  fraction   —    of  the    area 

of  the  whole  disk.     Hence,  M  being  the  mass  of  the  disk, 
the  mass  of  the  ring  is  -^  M.     The  moment  of  inertia  of 

*  By  calculus  method.  Let  p  be  mass  of  the  disk  per  unit  area.  The 
mass  of  a  narrow  ring  of  radius  r,  and  width  dr,  is  2  irr  •  dr  •  />,  and  its 
moment  of  inertia  about  the  axis  of  the  disk  is  2  irrdrp  •  r2. 


.-.  /=  ( 


MOMENT  OF  FORCE  101 

this  ring  about  an  axis  through  the  centre  perpendicular 
to  the  plane  of  the  disk  is  — ^  M  •  (n  — j  •  Summing 
up  for  all  values  of  n  from  1  to  JV,  we  get 


Hence,  supposing  N  indefinitely  gieat/>\ 
I=\MR*. 

It  is  readily  seen  from  the  note  in  §  73  that  the  moment 
of  inertia  of  the  circular  disk  about  a  diameter  is  ^  MR2. 

76.   Moment  of  Inertia  of  a  Right  Circular  Cylinder. — A 

right  circular  cylinder  may  be  divided  up  into  circular 
disks.  The  preceding  formula  applies  to  each  disk.  Add- 
ing the  moments  of  inertia  of  all  the  disks,  we  get  for  the 
moment  of  inertia  of  a  right  circular  cylinder  of  mass  M 
about  its  geometrical  axis 


It  will  be  shown  later  that  the  moment  of  inertia  of 
a  right  circular  cylinder  about  an  axis  parallel  to  its 
geometrical  axis,  and  at  a  distance  d  from  the  latter,  is 


77.   Radius  of  Gyration.  —  The  radius  of  gyration,  &,  of 
a  body  about  a  certain  axis  is  a  length  such  that,  if  the 


102  DYNAMICS 

whole  mass  were  supposed  concentrated  at  that  distance 
from  the  axis,  the  moment  of  inertia  would  be  unchanged, 
or  1=  Mk2.  For  a  rod  about  the  middle,  A?  =  -^  L2.  For 
a  circular  disk  about  the  axis  of  figure  k2  =  |  R2,  and  so  on. 

78.  Angular  Momentum.  —  If  a  body  have  an  angular 
velocity,  o>,  and  a  moment  of  inertia,  Z,  about  any  axis, 
the  product  Ico  is  called  the  angular  momentum  about  that 
axis.  Supposing  JTto  remain  constant,  the  rate  of  change 
of  Ico  is  I  multiplied  by  the  rate  of  change  of  &>,  i.e.  la. 
But  (7—  Tot)  .hence  the  moment  of  force  about  an  axis 
equals^  tlie  rate  of  change  of  angular  momentum  about 
•  ihat  axis.  *  It  "the-  moment  of  force  is  zero,  the  angular 
momentum  is  constant. 

Exercise  XVII.    Moment  of  Force  and  Angular  Acceleration 

Apparatus.  —  A  horizontal  circular  disk  of  wood  is  carried  by  a 
vertical  steel  axis  which  passes  through  the  centre  of  the  disk.  The 
steel  axis  is  mounted  in  cone  bearings,  the  bearing  points  being  steel 
needle-points  so  that  the  friction  is  reduced  to  a  minimum.  A  silk 
thread  is  wrapped  around  the  axis  and  passing  over  a  pulley  carries  a 
weight  the  position  of  which  can  be  observed  by  a  vertical  scale.  Lead 
cylinders  are  placed  on  the  disk  at  equal  distances  on  opposite  sides  of 
the  axis  so  as  to  increase  the  moment  of  inertia.  The  mass  of  each 
part  of  the  apparatus  is  stamped  on  it.  The  diameter  of  the  steel 
axis  may  be  measured  by  a  micrometer  caliper  (§3).  A  metre 
scale  and  a  simple  beam-compass  (§3)  are  used  for  measuring  the 
diameter  of  the  disk,  the  diameter  of  the  lead  cylinders  and  the  dis- 
tance of  the  latter  from  the  axis. 

Calculation  of  Time  of  Descent  of  Weight.  —  From  the  dimensions 
and  masses  of  the  various  parts  their  moments  of  inertia  about  the 
axis  of  rotation  are  calculated.  The  moment  of  the  force  exerted  by 
the  silk  thread  on  the  axis  is  deduced  from  the  tension  of  the  thread, 


MOMENT  OF  FORCE 


103 


and  the  radius  of  the  axis.  The  angular  acceleration  is  derived  from 
the  total  moment  of  inertia  and  the  moment  of  the  force.  The  time 
required  for  the  weight  to  descend  to  the  floor  is  then  calculated. 

Observation  of  Time  of  Descent  of  Weight.  —  The  disk  is  levelled  by 
means  of  a  spirit  level.     To  prevent  tangling  the  length  of  the  thread 


FIG.  43. 


should  be  such  that  the  weight  will  just  reach  the  floor.  To  prevent 
overlapping  of  the  windings  of  the  thread,  it  should  be  attached  to 
the  axis  at  a  point  slightly  higher  or  lower  than  the  point  at  which  it 
tends  to  wind  when  the  disk  is  turned.  With  this  adjustment  the 
thread  will  always  be  very  nearly  at  right  angles  to  the  axis.  The 


104  DYNAMICS 

upper  bearing  should  be  adjusted  until  the  axis  is  just  perceptibly 
loose  in  the  bearings. 

The  disk  is  then  rotated  and  the  thread  wound  on  the  axis  without 
overlapping  until  the  weight  rises  to  the  height  desired.  Then  exactly 
at  a  tick  of  the  clock  the  disk  is  released  and  the  number  of  whole 
seconds  and  fifths  of  a  second  before  the  weight  strikes  the  floor  is 
carefully  noted.  This  should  be  repeated  ten  times  and  a  strong 
effort  made  to  have  the  separate  determinations  as  independent  as 
possible.  The  average  of  these  is  taken  as  the  time  of  descent. 

(The  changes  of  energy  that  take  place  will  be  studied  in  Exer- 
cise XXIV.) 

DISCUSSION 

(a)  Meaning  and  proof  of  formulae  used. 

(6)  Is  it  necessary  that  the  axis  be  vertical  or  the  thread  horizontal 
or  is  it  sufficient  that  they  be  at  right  angles? 

(c)  Effect  of  using  a  thread  whose  thickness  is  comparable  with 
the  radius  of  the  axis. 

(d)  Effect  of  stretching  of  thread  and  overlapping  of  thread. 

(e)  Was  the  tension  just  equal  to  the  weight?     How  could  it  be 
found  exactly  ? 

(/)  Effect  of  bearings  not  being  quite  central. 

(#)  Is  there  any  reason  to  suppose  that  the  acceleration  is  not 
quite  constant? 

(A)  If,  after  the  weight  has  reached  the  floor,  the  disk  be  allowed 
to  continue  in  rotation  and  rewind  the  thread,  why  will  not  the  weight 
rise  to  its  original  height  ? 

(i)  What  would  be  the  effect  if  the  cylinders  were  made  of  the 
same  mass  as  before  but  of  twice  as  great  diameter  ? 

(_/)  An  iron  cylinder,  3  ft.  in  external  diameter  and  2  ft.  10  in. 
in  internal  diameter,  rolls  down  a  plane  20  ft.  long  inclined  at  30°  to 
the  horizontal.  What  linear  velocity  does  it  acquire  ? 

79.  Centre  of  Mass  of  a  Body.*  —  The  centre  of  mass  (or 
of  inertia)  is  a  point  of  great  importance  in  the  study  of 

*  The  part  of  this  section  preceding  the  definition  in  italics  may  (if 
thought  advisable)  be  omitted.  The  only  objection  to  this  is  that  it  will 
involve  the  assumption  that,  if  a  point  fulfil  the  condition  of  being  the 


MOMENT  OF  FORCE 


105 


a  group  of  particles  or  of  a  rigid  body.     The  centre  of 
mass  of  a  particle  ml  at  Pv  and  a  particle  w2  at  P2,  is  a 


point 


n 


such  that 


or 


If  a  third  particle  mz  at  P3  be  added,  the  centre  of  mass 
of  all  three  is  a  point  Q2  in  §1P3  such  that 


FIG.  44. 

and  so  on  for  any  number  of  particles,  and  hence  for  a 
rigid  body. 

The  centre  of  mass  can  be  more  briefly  defined  in  terms 
of  the  distances  of  the  particles  from  any  plane.  Let  the 
perpendiculars  from  Pv  P2,  P3,  — ,  Qv  Qv  •••,  on  any 
plane,  LMNR,  be  respectively  dv  d^  dy  •••,  dr  92,  •••,  and 

centre  of  mass  as  regards  distances  from  any  three  planes,  it  will  also 
fulfil  the  condition  as  regards  distances  from  any  other  plane.  The  proof 
of  this  proposition  would  require  more  knowledge  of  analytical  geometry 
than  we  assume  in  this  book. 


106  DYNAMICS 

let  the  feet  of  the  perpendiculars  be  Av  A2,  A&  •••,  B^ 
.Z?2,  •••,  respectively.  Produce  P^P^  anc^  ^-1^-2  ^°  meet  in 
0.  Then 

Q,P2  :  QlPl  -.OP.-OQ.iOQ,-  OPr 


or  (ml  4-  m2)51  =  m1d1  4  ?H2c 

It  can  be  shown  in  the  same  way  that 


Hence        m  +  w  4-  ^3  = 


The  process  can  evidently  be  extended  to  any  number 
of  particles,  and  so  to  a  continuous  body.  Hence  we  may 
define  centre  of  mass  thus  :  If  mv  ra2,  •••  are  the  respective 
masses  of  the  particles  constituting  a  lody  (or  group  of  par- 
ticles) of  total  mass  M,  and  if  the  respective  distances  of  these 
particles  from  any  plane  are  dv  d2  •••,  the  centre  of  mass  is  a 
point  whose  distance  from  the  plane  is 

_  m1d1  4-  ^2^2  4-  •••___  ^md 
m14w24-.-  M 

If  mv  mz"-  are  all  equal,  3  is  the  mean  of  the  distances 
dv  d2~-.  If  m  j,  w2  •••  are  unequal,  3  is  still,  in  a  sense, 
the  mean  distance,  but,  in  taking  the  mean,  the  distance 
of  each  particle  is  given  an  "  importance  "  measured  by 
the  mass  of  that  particle.  In  applying  the  definition  to 
a  plane  that  passes  through  the  group  of  particles  or  body, 


MOMENT  OF  FOECE  107 

distances  on  one  side  of   the   plane  must   be  considered 
positive  and  those  on  the  other  side  negative. 

If  dv  c?2,  etc.,  be  the  distances  of  the  particles  mv  m^ 
-etc.,  from  a  plane  that  passes  through  the  centre  of  mass, 
3  =  0  and  therefore  2mc?  =  0. 


80.  Coordinates  of  Centre  of  Mass.  —  The  position  of  the 
centre  of  mass  can  be  specified  definitely  by  its  distances 
(with  proper  signs)  from  any  three  planes  at  right  angles. 
If  in  any  problem  three  intersecting  lines  at  right  angles 
have  been  chosen  as  axes  of  rectangular  coordinates,  then 
the  distance  of  any  particle  m  from  the  plane  containing 
the  x  and  y  axes  is  z,  its  distance  from  the  plane  of  the  y 
and  2  axes  is  #,  and  its  distance  from  the  plane  of  the  x 
and  z  axes  is  y.  Hence,  if  x,  y,  ~z  be  the  coordinates  of 
the  centre  of  mass, 


_  _ 

~"'  y"  ~'      " 


81.  Centre  of  Mass  in  Simple  Cases.  —  When  a  homoge- 
neous body  has  a  geometrical  centre,  the  body  can  be  sup- 
posed divided  up  into  pairs  of  equal  particles,  each  pair 
lying  in  a  line  through  the  geometrical  centre  and  one  of 
the  pair  being  as  far  on  one  side  of  the  centre  as  the  other- 
is  on  the  other  side.  Hence  the  geometrical  centre  is  also 
the  centre  of  mass.  Hence  the  centre  of  mass  of  a  uniform 
rod,  a  uniform  circular  disk,  a  sphere,  a  right  circular 
cylinder,  a  parallelogram,  a  parallelepiped,  etc.,  is  in  each 
case  the  geometrical  centre. 

When  a  body  can  be  divided  up  into  a  number  of  parts 
the  masses  and  centres  of  mass  of  which  are  known,  the 


108 


DYNAMICS 


centre  of  mass  of  the  whole  body  can  be  found  by  means 
of  the  expressions  for  #,  y,  z  in  §  80. 

82.  Moment  of  Inertia  of  a  Body  about  any  Axis.  —  Let  / 
be  the  moment  of  inertia  of  a  body  about  an  axis  through 
A  perpendicular  to  the  plane  of  the  paper  and  /0  its  mo- 
ment of  inertia  about'  a  parallel  axis 
through  the  centre  of  mass  0.  Let 
m  be  a  particle  at  a  point  P.  From 
P  draw  perpendiculars  PA  and  PC 
to  the  two  axes.  Join  AC  and  from 
C  P  draw  a  perpendicular  PD  on  AC. 
Denote  CD  by  d.  Then 


D 
FIG.  45. 


7= 

OA2  _  2  P0.  OA  cos 

+  CA2$m  -2CA-  2(mPCcos  PCA). 

But  2(wP(7cos  PCA)  =  ^md  =  0  since  d  is  the  distance  of 
m  from  a  plane  through  the  centre  of  mass  C  and  perpen- 
dicular to  AC  (§  79).  Hence  if  we  denote  the  distance  AC 
between  the  two  axes  by  a  and  the  mass  of  the  body  by  M 

1=  J0  +  Ma2. 

As  an  example  of  the  usefulness  of  this  proposition  con- 
sider the  moment  of  inertia  of  a  circular  cylinder  of  mass 
Jf,  length  L,  and  radius  R  about  an  axis  through  the 
centre  of  the  cylinder  and  perpendicular  to  the  geometrical 
axis.  Divide  the  whole  length  of  the  cylinder  into  a  large 
number  of  equal  disks  by  planes  perpendicular  to  the  axis 
of  the  cylinder.  The  moment  of  inertia  of  a  disk  of  mass 
m  at  a  distance  x  from  the  centre  of  the  cylinder  is  (§  75) 

\  mR2  +  mx2. 


MOMENT  OF  FORCE  109 

The  sum  of  the  first  term  for  all  the  disks  is  ^  MR2.  The 
summation  of  the  second  term  has  been  performed  already 
in  finding  the  moment  of  inertia  of  a  rod  about  its  centre 
(§  72).  Hence 


83.  The  Conservation  of  Angular  Momentum.  —  The  angu- 
lar momentum  of  a  body  about  an  axis  is  constant  if  the 
moment  of  force  about  that  axis  is  zero  (§  78).  A  similar 
statement  may  be  made  with  regard  to  the  total  angular 
momentum  of  a  group  of  bodies  (e.g.  the  solar  system) 
about  an  axis  or  line  in  space,  even  if  the  bodies  have 
different  angular  velocities  and  are  not  at  fixed  distances 
from  the  axis.  The  moment  of  each  force  about  the  axis 
is  equal  to  the  angular  momentum  it  produces  per  second, 
and  if  the  sum  of  the  moments  of  all  the  forces  about  the 
axis  is  zero  the  total  change  of  angular  momentum  in  any 
time  is  zero.  Internal  forces,  that  is  forces  which  the 
bodies  exert  on  one  another,  are  made  up  of  pairs  of  equal 
and  opposite  forces  (Newton's  Third  Law),  and  have  there- 
fore zero  total  moment  about  any  axis.  If  the  external 
forces  that  act  on  the  system  have  also  zero  total  moment 
about  the  axis,  the  angular  momentum  about  the  axis  will 
be  constant.  This  principle  is  called  the  conservation  of 
angular  momentum.  The  following  exercise  will  illustrate 
a  particular  case. 

Exercise  XVIII.    Conservation  of  Angular  Momentum 

A  number  of  bodies,  rotating  about  an  axis  with  a  known  angular 
velocity,  move  to  greater  distances  from  the  axis  ;  find  by  calculation 
and  also  by  experiment  the  new  angular  velocity. 


110 


DYNAMICS 


A  vertical  steel  axis  supported  on  needle-points  carries  a  hori- 
zontal steel  cross-bar,  along  which  two  cylindrical  iron  blocks  can 
slide.  The  axis  is  set  into  rotation  by  a  thread  that  is  wrapped 


FIG.  46. 


around  the  axis,  and,  passing  over  a  pulley,  carries  a  weight.  By 
a  simple  device,  a  cord  that  restrains  the  sliding  blocks  is  released 
at  the  moment  when  the  thread  is  wholly  unwrapped  from  the  axis, 
or,  by  a  slight  change  in  the  arrangement,  the  thread  can  unwrap 
without  releasing  the  weights.  (The  cord  and  the  thread  loop  over 


MOMENT  OF  FOECE  111 

a  small  peg, />;  if  the  cord  be  outside  it  will  be  released;  if  inside, 
it  will  not  be  released.  If  the  cord  and  thread  be  placed  on  the  hook, 
k,  the  thread  will  not  be  detached  from  the  axis,  and  the  cord  will  be 
released  or  not  released,  according  to  whether  it  is  outside  or  inside.) 
The  initial  positions  of  the  blocks  are  fixed  by  pins  in  the  cross-bar, 
and  when  the  blocks  slide  out  they  are  arrested  in  definite  positions 
by  other  pins  in  the  cross-bar.  The  speed  of  rotation  of  the  axis  at 
any  time  is  ascertained  by  means  of  the  recording  disk,  pendulum, 
and  brush  described  in  Exercise  XIY  (p.  84).  Weigh  blocks,  rods, 
and  disk  before  setting  the  apparatus  up. 

Several  records  of  the  final  speed  should  be  obtained,  the  apparatus 
being  so  arranged  that  the  blocks  are  not  released.  For  each  record 
the  cord  is  wrapped  up  anew,  and  the  weight  is  allowed  to  descend 
from  the  same  height.  Again,  several  records  should  be  obtained 
after  the  blocks  have  been  released.  For  these  experiments  the 
thread  should  be  placed  on  the  peg,  p,  so  that  it  will  become  detached 
and  not  retard  the  motion  to  be  measured.  Each  record  should  be 
numbered  as  soon  as  obtained. 

The  distances  of  the  centres  of  the  blocks  from  the  centre  of  the 
axis  in  both  positions  of  the  blocks  should  be  measured  very  carefully 
by  means  of  the  beam-compass.  The  length  of  the  cross-bar,  the 
diameter  of  the  disk,  and  the  diameter  of  the  axis  should  also  be 
measured,  the  last  by  means  of  the  micrometer  caliper.  From  these 
measurements  the  total  moment  of  inertia,  both  before  and  after  the 
change  of  position  of  the  blocks,  can  be  calculated.  Taking  the 
initial  angular  velocity  as  known,  calculate  the  final  angular 
velocity. 

With  a  view  to  facilitating  Exercise  XXVIT,  which  is  a  continua- 
tion of  the  above,  record  the  total  distance  of  descent  of  the  weight, 
the  height  to  which  it  reascends  when  the  thread  is  not  detached  and 
the  moment  of  inertia  is  not  changed,  and  the  height  to  which  it 
reascends  when  the  moment  of  inertia  changes. 

DISCUSSION 

(a)  Sources  of  error. 

(6)  Meaning  and  proof  of  formulae  used. 

(c)  "  Centrifugal  force  "  of  the  blocks  in  both  positions. 


112  DYNAMICS 

I 

(d)  At  what  speed  of  rotation  (given  the  coefficient  of  friction) 
would  the  blocks  just  begin  to  slide  if  released? 

(e)  Does  the  friction  between  block  and  cross-bar  affect  the  result  ? 
(/)  How  could  data  be  obtained  for  drawing  a  curve  to  show  the 

way  in  which  the  speed  increases  with  distance  of  descent  of  the 
weight,  and  how  could  a  more  accurate  value  of  the  final  speed  be 
thus  obtained  ? 

(</)  Is  the  motion  of  the  blocks  affected  in  any  way  by  force  applied 
to  them  by  the  cross-bar  ? 

(A)  Why  does  water  in  a  wash-basin  rotate  as  it  runs  out  ? 

(i)  How  must  shrinkage  of  the  earth,  due  to  cooling,  affect  the 
length  of  the  day  ? 

84.  Velocity  and  Acceleration  of  Centre  of  Mass.  —  The 
velocity  of  a  particle  in  any  direction  is  the  rate  of  change 
of  its  distance  from  a  fixed  plane  perpendicular  to  that 
direction.  Let  the  distance  of  the  particles  mv  w2,  •••  from 
a  plane  be  dv  d%,  •••  respectively,  and  let  the  distance  of 
the  centre  of  mass  from  the  plane  be  8,  then  (§  79) 


After  a  short  interval  of  time,  T,  let  these  distances  be 

S\  <V>  «*,'...; 
then  (m1  +  mz+  •••)§'  =  m-^d^  -\-m2d2f  +  •••. 

If  the  first  equation  be  subtracted  from  the  second,  and 
both  sides  of  the  result  divided  by  T,  and  T  then  supposed 
indefinitely  short,  we  get 


v  being  the  velocity  of  the  centre  of  mass  in  the  direction 
in  which  the  distances  are  measured  and  vr  vv  •••  the  velo- 
cities of  w-p  m2,  •••  respectively,  in  the  same  direction. 
Applying  the  same  method  to  changes  of  velocity  we 


MOMENT  OF  FORCE  113 

can  show  that  if  a  be  the  acceleration  of  the  centre  of 
mass  in  any  direction,  and  «A,  a2,  ...  the  accelerations  of  the 
particles  mv  w2,  --  respectively, 

(^ml  -f-  w2  +  •  •«)«  =  m^  +  w2#2  +  «... 
These  two  equations  may  also  be  written  in  the  forms 
^m(v  —  F)  =  0  and  Sw(a  —  a)  =  0. 

Here  (v  —  v)  for  any  particle  m  is  its  velocity,  in  any 
direction,  relatively  to  the  centre  of  mass,  and  a  similar 
statement  applies  to  (a  — a). 

85.  Acceleration  of  the  Centre  of  Mass  of  a  Body  acted  on 
by  External  Forces.  —  The  forces  acting  on  a  group  of 
particles  may  be  classified  as  internal  and  external.  In- 
ternal forces  are  due  to  the  actions  and  reactions  between 
particles  themselves.  External  forces  are  the  forces  be- 
tween the  particles  and  outside  bodies.  The  internal 
forces  on  a  bridge  are  the  stresses  in  the  various  parts, 
the  external  forces  are  gravity,  wind  pressure,  reaction 
of  supports,  etc. 

Let  the  component,  in  any  given  direction,  of  the  result- 
ant of  the  external  forces  on  a  particle  m^  be  F±  and  let 
the  component  in  the  same  direction  of  the  resultant  of 
the  internal  forces  on  the  particle  be  /j ;  for  a  second 
particle  m2  let  the  corresponding  components  be  F2  and  /2 
and  so  on.  If  the  accelerations  in  that  direction  of  the 
various  particles  be  a1?  #2,  •••  respectively,  then  by  Newton's 
Second  Law 

Fl 


114  DYNAMICS 

If  a  be  the  acceleration  of  the  centre  of  mass  and  M  the 
whole  mass,  then  by  §  84 


By  Newton's  Third  Law  the  internal  forces  occur  in 
equal  and  opposite  pairs, 

.-.  2/=0. 
Hence  2F=  Ma. 


Thus  the  motion  of  the  centre  of  mass  in  any  direction  is  the 
same  as  if  all  the  mass  were  concentrated  at  the  centre  of 
mass  and  all  forces  were  transferred,  with  their  directions 
unchanged,  to  the  centre  of  mass.  Since  this  statement 
applies  to  any  group  of  particles,  it  applies  also  to  a  con- 
tinuous body. 

The  motion  of  the  centre  of  mass  of  a  body  is  not  affected 
by  internal  forces.  When  a  rocket  explodes,  the  position 
and  motion  of  the  centre  of  mass  are  not  affected  by  the 
explosion.  Similar  statements  may  be  made  with  refer- 
ence to  attracting  and  colliding  bodies. 

86.  Translation  and  Rotation.  —  If  the  linear  motion  of 
the  centre  of  mass  0  of  a  body  and  the  angular  motion 
of  the  body  about  O  are  known,  the  whole  motion  of 
the  body  is  known.  To  learn  how  any  number  of  forces 
applied  to  the  body  affect  its  motion,  it  is  only  necessary 
to  find  the  linear  acceleration  they  impart  to  O  and  the 
angular  acceleration  about  O  which  they  produce.  Each 
of  these  may  be  calculated  separately.  For  the  former  de- 
pends only  on  the  magnitudes  and  directions  of  the  forces 
and  not  on  their  points  of  application,  that  is,  on  their 
moments  about  O.  The  latter  depends  only  on  the 


MOMENT  OF  FORCE  115 

moments  of  the  forces  about  axes  through  C  and  is  un- 
changed if  additional  forces  be  applied  to  0  to  keep  it  at 
rest,  for  such  forces  would  have  no  moments  about  (7. 
Thus  we  may  consider  translation  of  the  centre  of  mass 
and  angular  acceleration  about  the  centre  of  mass  as  the 
two  independent  effects  of  the  forces  applied  to  a  body. 

87.  D'Alembert's  Principle.  —  The  equation  proved  in 
§  85  is  very  important.  Written  in  the  form  ^F — 2ma  =  0, 
it  is  sometimes  called  D'Alembert's  Principle.  Since  ma 
equals  the  force  that  would  give  the  mass  m  the  accelera- 
tion a,  it  was  sometimes  called  the  "  effective  force  "  act- 
ing on  the  particle  and  —  ma  was  called  the  "  reversed 
effective  force."  1LF  is  the  sum  in  a  certain  direction  of 
the  components  of  the  external  forces,  and  —  ^ma  is  the 
sum  in  the  same  direction  of  the  components  of  the  "  re- 
versed effective  forces,"  and  since  these  sums  of  compo- 
nents added  together  equal  zero,  the  whole  of  the  external 
forces  and  the  whole  of  the  reversed  effective  forces  may 
be  considered  as  a  system  in  equilibrium.  Hence  D'Alem- 
bert's  Principle  is  usually  stated  thus  :  "  The  external 
forces  with  the  reversed  effective  forces  of  a  system  of 
particles  constitute  together  a  system  of  forces  in  equi- 
librium." 

Exercise  XIX.    Friction.    D'Alembert's  Principle 

Four  rectangular  blocks  of  the  same  wood  with  plane,  clean,  freshly 
sandpapered  surfaces  are  placed  on  one  another  as  in  Fig.  47,  the 
lower  one  resting  on  another  block  of  like  material  and  finish. 
What  force,  F,  applied  horizontally  to  one  of  them,  for  example  the 
third  from  the  top,  will  pull  it  free  of  the  others  ? 

Let  the  masses  be  Mv  Mv  M&,  M4  respectively  and  the  respective 
accelerations  av  a2,  a3,  a4.  The  reader  will  find  little  difficulty  in  show- 


116 


DYNAMICS 


ing  that  a4  must  be  zero.     The  force  P  must  be  such  as  to  produce 
in  M3  an  acceleration,  a3,  greater  than  the  acceleration,  a2,  of  M2. 
If  P  be  the  pressure  between  two  of  the  blocks  and  /x  the  coefficient 
of  static  friction,  the  greatest  horizontal  force  one 
can  exert  on  the  other  is  Pp.    Hence  the  greatest 
force    possible    between   Ml   and    M2   is    M^p. 
If  no  slipping  takes  place  between  Ml  and  /T/2, 
the  force  between  them  will  be  less  than  M^p,  say 
Mtffji—  8,  8  being  either  zero  or  a  positive  quantity. 
We  can  by  D'Alembert's  Principle  write  down 
the  equations  for  the  following  systems  :  (1)  J/L 
alone,  (2)  Ml  and  M2  together,  (3)  Mv  M2,  and 
M3  together. 

-  8  -  M,C(I  =  0.       (1) 

fr  +  M2a2)  =  0.        (2) 


-  (Mp,  +  M2a2  +  M3a3)  =  0.       (3) 
Subtracting  (1)  from  (2),  we  get 

M2g/ji  +  8  —  M2a2  =  0. 

Hence  a2  ^  gp..     Hence  when  slipping  between 
z  and  M3  just  takes  place,  a3^  gp. 
Subtracting  (2)  from  (3),  we  get 
P  -  (2  Ml  +  2  M2  +  M3)  ffft,  -  M3a3  =  0. 


The  student  should  also  solve  the  problem  by 
writing  down  Newton's  Second  Law  for  each  one 
of  the  blocks.  This  will  make  it  clear  to  him 
that  D'Alembert's  Principle  is  equivalent  to 
Newton's  Laws  of  Motion. 

To  find  the  least  force,  F,  that  will  cause  slip- 
ping between  M2  and  M3,  attach  a  calibrated 
spring  to  M3  and  fasten  the  other  end  of  the 
spring  to  an  upright  by  means  of  a  string  of  adjustable  length  (a 
bracket  screwed  to  the  table  will  do  for  the  upright,  and  the  string 
may  be  passed  through  a  hook  in  the  bracket).  The  spring  should  be 
horizontal ;  it  should  be  attached  to  the  middle  of  the  end  of  My,  and 


MOMENT  OF  FORCE  117 

its  direction  should  be  perpendicular  to  the  end  of  My  Several  trials 
will  be  necessary  in  order  to  find  the  length  to  which  the  spring  must 
be  stretched  so  that  when  M3  is  released,  slipping  takes  place  between 
it  and  M2. 

The  following  two  simple  devices  will  greatly  facilitate  the  obser- 
vations. A  lever  attached  to  the  end  of  M4  and  carrying  a  stud  that 
passes  through  a  screw-eye  in  the  end  of  M3  is  convenient  for  releas- 
ing M3  promptly  and  uniformly.  A  small  paper  index  fastened  by  a 
pin  to  the  side  of  M3  and  pressing  against  a  pin  on  the  side  of  M2 
will  be  displaced  if  any  slipping  between  M2  and  M3  takes  place,  and 
is  convenient  for  recording  any  actual  slip.  Without  it  a  slight  back- 
ward slip  of  M2  at  start  of  M3,  being  compensated  by  an  equal  for- 
ward slip  when  M3  stops,  would  sometimes  pass  unnoticed.  When 
the  desired  adjustment  of  the  length  of  the  spring  has  been  obtained, 
the  length  of  the  spring  must  be  measured  with  great  accuracy  by 
means  of  the  beam-compass  or  mirror-scale. 

The  coefficient  of  friction  between  M3  and  M4  is  found  from  the 
force  (applied  by  the  spring)  necessary  to  cause  M3  to  move  on  M4. 
If  the  surfaces  of  the  blocks  have  been  freshly  sandpapered,  the 
coefficients  of  friction  between  the  blocks  will  be  found  appreciably 
equal.  To  obtain  the  value  of  the  coefficient  as  accurately  as  possible, 
find  the  friction  between  M3  and  M4  when  all  four  blocks  are  in  posi- 
tion, then  when  Ml  has  been  removed,  and  finally  when  Ml  and  M2 
have  been  removed.  By  plotting  the  various  values  of  the  friction 
against  the  corresponding  values  of  the  pressure  of  M3  on  M4,  a  very 
reliable  estimate  of  the  coefficient  of  friction  should  be  obtained  and 
at  the  same  time  the  constancy  of  the  ratio  of  friction  to  pressure  will 
be  tested.  Other  arrangements  of  these  positions  of  the  blocks  may 
be  tried  if  time  permit.  (Instead  of  a  spring  in  the  preceding  exer- 
cise, a  cord  that  passes  over  a  pulley  and  carries  a  pan  and  weights 
might  be  used.  This  would  complicate  the  calculation,  owing  to  the 
inertia  of  the  pan  and  weights  and  the  friction  of  the  pulley,  and  on 
the  whole  gives  less  satisfactory  results.) 

DISCUSSION 
(a)   If  sliding  of  M2  were  prevented,  what  force  would  just  start 


118  DYNAMICS 

(&)    What  force  would  just  cause  M3  to  slide  on  M"4? 

(c)  Try  to  state  in  general  language  why  the  force  required  to 
pull  M3  free  from  M2  and  M4,  as  in  the  exercise,  is  greater  than  that 
calculated  in  (a). 

(a?)    Why  does  not  M4  move  ? 

(e)  Has  Ml  any  tendency  to  slide  on  M2  in  the  experiment  and 
why? 

(/)  In  what  respect  would  the  solution  of  the  problem  of  the  ex- 
ercise be  different  if  the  coefficients  of  friction  between  the  various 
pairs  of  blocks  were  different?  w 

(</)  At  what  angle  to  the  horizontal  would  the  platform  on  which 
M4  rests  have  to  be  tilted  so  that  M4  would  slide  downward  ? 

(h)  If  the  platform  were  tilted  as  in  (g),  between  what  pair  of 
blocks  would  sliding  first  begin? 

(i)  If  the  platform  were  tilted,  but  not  sufficiently  to  produce 
sliding,  what  amount  of  force  would  be  called  into  play  between 
each  pair  of  blocks  ? 

REFERENCES 

Gray's  "  Treatise  on  Physics,"  Vol.  I,  Chapter  IV. 
Macgregor's  "  Kinematics  and  Dynamics,"  Part  II,  Chapters  V  and 
VI. 


CHAPTER  VII 

RESULTANT  OF  FORCES.     EQUILIBRIUM 

88.  Resultant  of  Forces  acting  on  a  Body.  —  The  forces 
acting  on  a  particle  may  be  reduced  to  a  single  equivalent 
force  called  the  resultant  of  the  forces.  When  forces  act 
at  different  points  of  a  body,  they  may  in  certain  cases  be 
reduced  to  a  single  equivalent  force.  In  other  cases  no 
single  force  will  produce  the  same  results  as  the  actual 
forces.  For  example,  suppose  the  centre  of  mass  of  a  body 
has  a  linear  acceleration  in  a  certain  direction,  while  the 
body  has  an  angular  acceleration  about  an  axis  in  that 
direction  through  the  centre  of  mass.  A  little  considera- 
tion will  show  that  no  single  force  could  produce  exactly 
the  same  linear  and  angular  accelerations. 

The  resultant  of  the  forces  acting  on  a  body  is  the  single 
force  or  the  simplest  set  of  forces  that  will  give  the  body 
the  same  accelerations,  linear  and  angular,  as  the  actual 
forces  produce.  Hence  we  get  the  following  conditions 
that  the  resultant  must  satisfy : 

(1)  The  resultant  must  give  the  centre  of  mass  of  the 
body  the  same  linear  acceleration  in  any  direction  as  the 
actual  forces  produce.  Hence  the  component  of  the  resultant 
in  any  direction  must  equal  the  sum  of  the  components  of  the 
actual  forces  in  that  direction.  It  is,  however,  not  neces- 
sary to  consider  all  directions  through  the  centre  of  mass. 
For  an  acceleration,  a,  in  any  direction  is  equivalent  to 

119 


120 


DYNAMICS 


three  accelerations,  av  «2,  #3,  in  directions  at  right  angles, 
such  that  a2  =  a?  +  a*  +  a32. 

A  force  that  will  cause  these  component  accelerations  will 
give  rise  to  a  and  will  satisfy  this  condition.  Hence  it  is 
sufficient  to  consider  three  directions  at  right  angles. 

(2)  The  resultant  must  give  the  body  the  same  angular 
acceleration  about  any  axis  as  the  actual  forces  produce. 
Hence  the  moment  of  the  resultant  about  any  axis  must  equal 
the  sum  of  the  moments  of  the  actual  forces  about  that  axis. 
For  a  reason  precisely  similar  to  that  stated  in  (1),  it  is 
only  necessary  to  consider  three  rectangular  axes  through 
a  point.  Moreover,  it  is  not  actually  necessary  to  con- 
sider rectangular  axes  through  all  points.  For  (§  86)  the 
two  independent  motions  of  a  body  are  linear  motion  of 
the  centre  of  mass  and  angular  motion  about  an  axis 
through  the  centre  of  mass.  Hence  it  is  sufficient  to 
consider  rectangular  axes  through  the  centre  of  mass 
only. 

89.  Resultant  of  Two  Parallel  Forces. —  (1)  Let  the  forces 
P  and  Q  be  in  the  same  direction  and  let  their  points  of 

application  be  A  and  B 
respectively.  Consider 
a  single  force  R=P+Q 


parallel  to  P  and  Q 
and  applied  at  a  point 
C  in  AB  such  that 
P-AC=Q-BC.  We 
shall  show  that  the  force 
R  satisfies  the  condi- 
tions for  being  the  re- 


B 


Qv 


FIG.  48. 


RESULTANT  OF  FOECES  121 

sultant  of  P  and  Q.  Since  R  is  in  the  same  direction 
as  P  and  Q  and  equals  their  sum,  the  component  of  R 
in  any  direction  equals  the  sum  of  the  components  of 
P  and  Q  in  that  direction.  Hence  R  satisfies  the  first 
condition  of  §  88. 

Let  an  axis  perpendicular  to  the  plane  of  P  and  Q  cut 
that  plane  in  0,  and  let  OA',  OB',  00'  be  perpendiculars 
from  0  on  P,  Q,  and  R  respectively. 

=(P+  Q)OC' 

=  P  •  00'  +  Q  -  00' 

=  P(OA'  -  A'  0")  +  Q(OB'  +  B'  Or) 

.  B'O'  -  P  -  A'O'. 


If  0  be  -the  angle  between  AB  and  A'B', 

Q.B'O'-P.  A'V  =  (Q  •  BO-  P.  AO)  cos  0  =  0. 
.-.  R-00'  =P.OA'  + 


Hence  the  moment  of  R  about  this  axis  equals  the  sum 
of  the  moments  of  P  and  Q.  It  is  readily  seen  that  the 
same  is  true  for  any  axis  perpendicular  to  the  first,  that  is, 
for  any  axis  in  a  plane  parallel  to  the  plane  of  P,  Q,  and  R. 
Hence  R  satisfies  the  second  condition  of  §  88. 

Hence  R  is  the  resultant  of  P  and  Q. 

The  point  of  application  of  the  resultant,  (7,  is  some- 
times called  the  "centre"  of  the  parallel  force.  Its  posi- 
tion is  evidently  independent  of  the  actual  direction  of 
the  parallel  forces  and  would  not  be  changed  if  they  were 
turned  in  some  other  direction,  while  still  remaining 
parallel. 

(2)  Let  the  forces  P  and  Q  be  in  opposite  directions 
and  let  P  be  >  Q.  Consider  a  force  R  =  P  —  Q  parallel 


122 


DYNAMICS 


to  P  and  Q  and  applied  at  a  point  0  in  BA  produced 
beyond  A  such  that  P  •  AC=  Q  •  BO.  R  will  evidently 
satisfy  the  first  condition  required  of  the  resultant  of 
p  P  and  Q. 

Let  an  axis  perpen- 
dicular ,to  the  plane  of 
P  and  Q  cut  that  plane 
in  <9,  and  let  OA  ', 


r  be   perpendiculars 
to  P,  (),  and  R  respec- 
FIG.  49.  tively.     Then 

R.  0(7'=  (P-  <))0(7' 

=  P(6U'  +  A'  C")  -  #(  OB1  +  .#'  C") 
=  P  -  OA'  -  Q  •  OB'  +  P  •  ^'(77  -  C  •  -#'#'. 
If  6  be  the  angle  between  AC  and  J/  (7', 
p.A'C'-Q.  B'C'  =  (P  -  AC-  Q  -  JBCT)  -  cos  (9  =  0. 
.-.  R 


Hence  the  moment  of  R  about  the  axis  equals  the  sum 
of  the  moments  of  P  and  Q  and  the  same  is  evidently 
true  for  any  axis  parallel  to  the  plane  of  P  and  Q. 

Hence  R  is  the  resultant  of  P  and  Q. 

Since  P-AO=Q.£0 


it  follows  that  the  less  the  difference  between  P  and  Q 
the  farther  0  is  from  A  or  B. 


RESULTANT  OF  FOECES  123 

(3)  Let  P  and  Q  be  equal  and  opposite. 

If  in  the  preceding  we  suppose  P  =  §,  then  R  =  0  and 
A0=  oo  or  the  resultant  in  a  zero  force  at  an  infinite  dis- 
tance. This,  however,  is  not  a  force  having  any  real 
existence  ;  it  is  merely  a  mathematical  fiction.  No  single 
force  can  be  found  that  is  equivalent  to,  i.e.  the  resultant 
of,  a  pair  of  equal  and  opposite  parallel  forces.  Such  a 
pair  of  forces  is  called  a  couple. 

Parallel  forces  in  the  same  direction  are  sometimes 
called  like  forces;  those  in  opposite  directions,  unlike 
forces. 

90.  Resultant  of  a  Number  of  Parallel  Forces.  —  Let 
Pj,  P2,  •••  be  parallel  forces.  If  they  are  all  in  the  same 
direction,  Pl  and  P2  are  equivalent  to  a  single  force  R± 
that  may  be  found  as  in  §  89,  Rl  and  P3  are  equivalent  to 
a  single  force  _R2,  and  so  on.  The  final  result  will  be  a 
single  force  R  which  is  therefore  the  resultant.  Evidently 


If  the  forces  are  not  all  in  one  direction,  those  in  one 
direction  are  equivalent  to  a  single  force  Rl  and  those  in 
the  opposite  direction  to  a  single  force  Rz.  If  Rl  and  R^ 
are  unequal,  their  resultant  is  a  single  force  R  =  2P.  If 
Rl  and  R2  are  equal,  that  is  if  2P  =  0,  their  resultant  is 
a  couple  (or  else  zero). 

From  the  way.  in  which  R  is  found  above  it  is  evident 
that  its  point  of  action  is  independent  of  the  actual  direc- 
tion of  the  parallel  forces. 

A  second  method  is  to  employ  the  principle  that  the 
moment  of  the  resultant  about  any  axis  must  equal  the  sum 
of  the  moments  of  the  components  (§  88).  Let  the 


124 


DYNAMICS 


distances  of  the  forces  from  an  axis  perpendicular  to  them 
be  pv  py  •••  respectively.  If  2P  is  not  zero,  the  resultant 
is  a  single  force  equal  to  2P  and  its  distance,  r,  from  the 
axis  is  given  by  Rr  —  SPjt?.  If  the  same  method  be  applied 
to  a  second  axis,  perpendicular  to  the  forces  and  to  the  first 
axis,  it  will  give  the  distance  of  the  resultant  from  the 
axis  and  these  two  distances  will  give  the  line  of  action  of 
R.  If  all  the  forces  are  in  one  plane,  the  resultant  is  in 
that  plane  and  it  will  be  sufficient  to  consider  moments 
about  a  single  axis  perpendicular  to  that  plane. 

A  third  method  gives  the  point  of  action,  (7,  of  R  if  the 
points  of  action  of  the  forces  are  known.     Let  the  dis- 
tance of  C  from  any  plane  be  #,  and  let 
the  distances  of  the  points  of  action  of 
the  forces  from  that  plane  be  xv  xv  -•• 
respectively.     Since  the  position  of  0  is 
independent  of  the  actual  direction  of 
the  parallel  forces,  we  may  suppose  them 
all  turned  parallel  to  the  plane  referred 
to.     Now  take  moments  about  any  axis, 
A,  lying  in  the  plane  and  perpendicular 
to  the  new  direction  of  the  forces.    The 
distance  of  the  new  line  of  action  of  Pl 
from  that  axis  is  xv  that  of  P2  is  #2,  and  so  on.      The 
distance  of  R  from  the  axis  is  x.     Hence,  Rx  =  2P#,  and 
this  gives  the  distance  of  0  from  the  plane.     The  same  is 
true  for  any  plane.      Hence,  taking  three  planes  at  right 
angles,  and  denoting  distances  from  it  by  #,  ?/,  and  z  re- 
spectively, we  have  for  the  position  of  0 

]S  r^T  *S  fy/  ^j  ~P  y 

~     -Ft  '    y  ~     T>  '  T>  * 

JK  Jtti  Jf> 


RESULTANT  OF  FORCES  125 

If  2P  =  0,  the  second  and  third  methods  fail ;  but  in 
that  case  \ve  may  first  omit  one  of  the  forces  and  find  the 
resultant  of  the  others.  This  with  the  omitted  force  will 
form  a  couple,  which  is,  therefore,  the  resultant  of  all  the 
forces. 

Exercise  XX.    Composition  of  Parallel  Forces 

(1)  Forces  in  the  same  direction.  —  A  very  light  framework,  con- 
sisting of  two  wooden  rods  crossed,  is  supported  in  front  of  a  cross- 
section  board  by  an  axis  that  passes  through  the  intersection  of  the 
rods.     Masses  are  suspended  from  the  ends  of  the  rods  by  means 
of  long  threads  so  that  they  hang  below  the  board.     Each  thread  is 
attached  to  the  rod  by  means  of  a  thumb-tack,  and  hangs  over  the 
end  of  the  rod,  so  that  the  point  of  application  of  the  force  is  sharply 
defined  by  an  edge  of  the  end  section.     (There  is  a  small  projection 
on  each  end  of  the  rear  face  of  the  front  rod,  and  the  thread  is 
attached  to  it,  so  that  it  may  hang  clear  of  the  other  rod  and  all  the 
strings  may  be  in  the  same  plane.) 

One  of  the  masses  is  a  scale  pan  carrying  weights.  The  weights 
are  adjusted  until  the  framework  hangs  in  equilibrium,  with  all  the 
threads  at  considerable  distances  from  the  axis  of  support.  Instead 
of  the  scale  pan  a  calibrated  spring  may  be  used.  The  resultant  is 
then  found  by  the  first  method  of  §  90,  the  forces  being  taken  in 
order  around  the  quadrilateral  formed  by  their  points  of  action. 
The  second  and  third  methods  are  then  applied  to  find  the  position 
and  point  application  of  the  resultant.  •  All  the  results  should  be 
represented  in  diagrams. 

(2)  Forces  in  opposite  directions.  —  Forces  acting  vertically  upward 
are  produced  by  calibrated  springs  attached  to  pegs  at  the  top  of  the 
board.     Care  should  be  taken  that  the  board  is  properly  levelled  and 
that  the  lines  of  action  of  the  springs  are  truly  vertical.     A  diagram 
is  drawn  as  before  and  the  resultant  found  by  all  three  methods. 


126  DYNAMICS 

DISCUSSION 

(a)  How  would  the  results  have  differed  if  the  forces  had  been 
at  some  inclination  to  the  vertical,  the  same  for  all? 

(&)  What  should  be  the  relative  position  of  the  point  of  action 
of  the  resultant  and  the  supporting  knife-edge  ? 

(c)  Compare  the  methods  of  finding  the  resultant  of  parallel  forces 
in  the  same  direction  and  the  methods  of  finding  the  centre  of  mass 
of  a  number  of  particles. 

(e?)  If  parallel  forces  proportional  to  their  masses  be  applied  to  a 
number  of  particles,  where  will  the  point  of  action  of  their  resultant  fall? 

(e)  How  would  the  framework  begin  to  move  if  one  thread  broke  ? 

(/)  Suppose  in  (2)  the  supporting  knife-edge  were  not  used. 
What  would  be  the  initial  motion  if  one  thread  broke  ? 

91.  Couples.  —  The  moment  of  a  couple  about  an  axis 
perpendicular  to  the  couple  is  constant,  that  is,  independent 
of  the  position  of  the  axis.  For  let  0  be  the  projection  of 
any  axis.  Then  if  0  be  between  P  and  Q,  the  moment 
of  the  couple  about  0  is  P  •  OA  +  Q  -  OB  =  P  •  AB.  If 

0   be    not    between   P    and 


Q 


P.OA-Q.OB=P*AB.    Hence 
the  moment  of  the  couple  is  the 
product  of  either  force  by  the  dis- 
«0  tance  between  the  forces  and  is 


the  same  about  all  axes  at  right 
angles  to  the  plane  of  the  couple. 

If  the  resultant  of  all  the  forces  acting  on  a  body  is  a 
couple,  the  motion  of  the  centre  of  mass  will  not  change 
(§  85),  but  the  couple  will  produce  an  angular  acceleration 
about  the  centre  of  mass  proportional  to  the  moment  of 
the  couple.  Hence  it  follows  that  all  couples  in  a  plane 
are  equivalent  if  they  have  equal  moments. 


RESULTANT  OF  FORCES  127 

Any  length  perpendicular  to  a  couple  and  proportional 
to  the  moment  of  the  couple  may  be  used  to  represent 
the  couple  and  is  called  the  axis  of  the  couple. 

It  is  shown  in  more  advanced  works  that  the  resultant 
of  any  number  of  forces  applied  to  a  body  is  a  single 
force  and  a  couple  the  axis  of  which  is  parallel  to  the 
line  of  action  of  the  force.  In  particular  cases  either  the 
force  or  the  couple  may  be  zero.  (Macgregor's  "  Kine- 
matics and  Dynamics,"  §§  479-482.) 

92.  Centre  of  Gravity.  —  If  to  the  particles  of  a  body 
parallel  forces,  proportional  to  the  masses  of  the  particles 
and  all  in  the  same  direction,  be  applied,  the  point  of 
action  of  the  resultant  will  be  at  the  centre  of  mass.  For 
if  P  be  the  force  on  a  particle  m,  P  —  Jem  where  k  is  some 
constant ;  and  if  R  be  the  resultant  force  and  M  the 
whole  mass,  R  =  k  •  M. '.  To  find  the  point  of  action  of  the 
resultant  we  substitute  these  values  in  the  formulae  of 
§  90.  On  doing  so  the  k  cancels  out  and  we  get  formulse 
identical  with  those  defining  the  centre  of  mass  in  §  80. 

The  weights  of  the  particles  of  a  body  are  forces  pro- 
portional to  the  masses  of  the  particles  and  they  are  prac- 
tically parallel,  provided  the  body  be  of  moderate  size. 
The  point  of  action  of  the  resultant  is  called  the  centre  of 
gravity  of  the  body.  It  follows  from  the  above  that  the 
centre  of  gravity  of  a  body  coincides  with  its  centre  of  mass. 

It  should,  however,  be  noted  that,  in  general,  only  a 
body  of  moderate  dimensions  can  be  said  to  have  a  centre 
of  gravity,  although  bodies  of  certain  particular  forms 
have  centres  of  gravity,  no  matter  what  their  magnitudes. 
Every  body  has  a  centre  of  mass. 


128  DYNAMICS 

93.  Equilibrium  of  a  Body.  —  A  body  is  in  equilibrium 
when  its  motion  is  constant,  that  is  when  the  linear  velocity 
of  its  centre  of  mass  is  constant  and  its  angular  velocity 
about  any  axis  is  constant.     If  no  forces  acted  on  a  body, 
it  would  be  in  equilibrium ;    but  a  body  may  also  be  in 
equilibrium  when  forces  act  on  it.     They  must,  however, 
satisfy  certain  relations  called  the  conditions  of  equilibrium. 

Given  that  a  body  is  in  equilibrium  we  may  conclude  that 

(1)  The  sum  of  the  components,  in  any  direction,  of  the 
forces  acting  on  the  body  is  zero,  since  the  linear  accelera- 
tion of  the  centre  of  mass  is  zero. 

(2)  The  sum  of  the  moments,  about  any  axis,  of  the  forces 
acting  on  the  body  is  zero,  since  its  angular  acceleration 
about  any  axis  is  zero. 

These  conditions  may  be  briefly  stated  thus  : 

0  in  any  direction. 
=  0  about  any  axis. 
Conversely,  if  both  these  conditions  are  satisfied,  the 
body  is  evidently  in  equilibrium. 

94.  Experimental  Method  of  finding  the  Centre  of  Gravity. 
—  A  body  suspended  by  a  cord  is  acted  on  by  two  forces, 
gravity,  acting  vertically  at  the  centre  of  the  body,  and 
the  tension  of  the  cord.     For  equilibrium  these  two  forces 
must  be  equal  and  opposite  and  in  the  same  line.     Hence 
the  centre  of  gravity  lies  in  the  line  of  the  cord  produced. 
By  suspending  a  disk  by  a  cord  attached  in  succession  to 
two  different  points  in  the  margin  of  the  disk  and  finding 
the  point  of  intersection  of  the  lines  in  the  disk  that  coin- 
cide in  succession  with  the  line  of  the  cord,  the  centre  of 
gravity  of  the  disk  may  be  located. 


RESULTANT  OF  FORCES 


129 


Exercise  XXI.    Equilibrium  of  a  Body 

A  disk  of  wood  is  suspended  in  a  vertical  plane  in  front  of  a  verti- 
cal cross-section  board  by  means  of  a  knife-edge  driven  into  the  board 


FIG.  52. 


and  passing  loosely  through  a  hole  in  the  disk.     Forces  are  applied  to* 
the  disk  by  springs  attached  as  in  Exercises  XI  and  XII  and  by  cords 


130  DYNAMICS 

hanging  vertically  from  the  disk  and  carrying  weights.  The  springs 
should  be  calibrated  before  they  are  used.  From  previous  experience 
it  will  be  obvious  that  two  points  on  the  calibration  curve  will  suffice. 

Since  the  force  of  gravity  on  the  disk  must  be  taken  into  account, 
the  position  of  the  centre  of  gravity  of  the  disk  should  be  determined 
by  the  method  of  §  94  before  the  disk  is  mounted  in  front  of  the  cross- 
section  board.  In  setting  the  apparatus  up  see  that  the  lines  of  all 
the  forces  are  at  considerable  distances  from  the  axis. 

The  distance  of  each  force  from  the  axis  may  be  obtained  by 
stretching  a  long  thread  very  accurately  along  the  line  of  action  of 
the  force.  The  inclination  of  the  thread  to  the  (positive)  horizontal 
should  be  measured  by  a  protractor.  The  length  of  the  springs  may  be 
found  by  the  beam  compass  or  mirror  scale.  From  these  data  calculate 
the  sum  of  the  moments  of  the  various  forces  (including  gravity)  about 
the  axis  of  rotation.  The  accuracy  of  the  result  should  be  estimated 
by  the  percentage  difference  of  positive  and  negative  moments. 

The  magnitude  R  and  direction  0  of  the  reaction  of  the  axis  on  the 
disk  can  be  found  by  equating  to  zero  the  sum  of  the  components  of 
all  the  forces  (including  the  unknown  reaction  of  the  axis)  first  in 
the  horizontal  direction  and  then  in  the  vertical  direction, 
.Rcos0  +  ^Fcosa  =  0 
R  sin  0  +  2F  sin  a  =  0. 

The  reaction  of  the  axis  may  also  be  found  experimentally  by  remov- 
ing the  axis  and  replacing  it  by  a  peg  occupying  the  same  hole  in  the 
disk  and  then  attaching  to  the  peg,  by  a  small  ring,  a  cord  that  passes 
over  a  pulley  and  carries  a  scale  pan.  Weights  are  placed  in  the  scale 
pan  and  the  position  of  the  pulley  adjusted  until  the  disk  conies  to  rest 
in  its  former  position. 

DISCUSSION 

(a)    Graphical  method  of  finding  the  reaction  of  the  axis. 
(&)   Motion  of  the  centre  of  gravity  of  the  disk  if  the  axis  were 
suddenly  removed. 

(c)  Angular  motion  of  the  disk  if  the  axis  were  suddenly  removed. 

(d)  Motion  of  the  disk  if  the  vertical  cord  were  to  break. 

(e)  How  would  the  position  of  the  disk  alter  if  board  and  disk  were 
turned  into  a  horizontal  plane,  the  forces  applied  remaining  unchanged? 


RESULTANT  OF  FORCES  131 

95.  Special  Cases  of  Equilibrium.  —  The  following  special 
cases  of  equilibrium  are  important : 

(1)  Two  forces.     For  equilibrium  they  must  evidently 
be  equal  and  opposite  and  act  in  the  same  line. 

(2)  Three  parallel  forces.     Any  one  must  be  equal  and 
opposite  to  the  resultant  of  the  other  two.     Hence  all 
three  must  be  in  the  same  plane. 

(3)  Three  non-parallel  forces  in  the  same  plane.     The 
moments  of  any  two  about  the  point  in  which  their  lines 
intersect  are  zero.     Hence  the  moment  of  the  third  about 
that  point  is  zero,  that  is,  its  line  of  action  also  passes 
through  that  point.     Hence  all  three  act  through  a  single 
point,  and  any  one  is  equal  and  opposite  to  the  resultant 
of  the  other  two. 

(4)  Three  forces  cannot  in  any  case  produce  equilibrium 
unless  they  act  in  the  same  plane.     For  consider  moments 
about  any  line  that  intersects  the  lines  of  two  of  the  forces. 
The  moments  of  these  two  about  any  such  line  are  zero. 
Hence  the  moment  of  the  third  about  it  is  also  zero.     That 
is,  the  third  must  either  be  parallel  to  every  such  line 
(which  is  impossible)  or  it  must  intersect  any  such  line, 
and  the  latter  can  evidently  only  be  true  if  the  lines  of 
action  of  all  the  forces  are  in  the  same  plane. 

Exercise  XXII.    Equilibrium  of  a  Body 

(1)  From  a  uniform  iron  rod  of  mass  m  and  length  /,  a  mass  M  is 
suspended  at  a  distance  h  from  one  end  of  the  rod,  and  the  rod  is  sup- 
ported by  a  horizontal  force,  Fv  applied  to  the  lower  end  and  a  force, 
Fy  applied  to  the  upper  end  at  an  angle  of  45°  with  the  horizontal. 
Find  Fl  and  jF2,  and  the  inclination  of  the  rod  to  the  horizontal. 

This  problem  is  to  be  solved  theoretically,  and  the  result  tested  experi- 
mentally. For  the  experimental  work  the  rod  is  suspended  in  front  of 


132  DYNAMICS 

the  cross-section  board  by  two  springs  which  apply  the  forces  Fl  and 
F2.  The  springs  are  attached  to  small  hooks  in  the  end  of  the  rod, 
the  other  ends  of  the  springs  being  borne  by  pegs  inserted  in  the  cross- 
section  board.  The  cord  that  sustains  M  may  be  fastened  to  the  rod 
by  a  slip-noose  that  binds  on  the  rod,  or  it  may  pass  through  a  hole  in 
the  rod.  The  tensions  of  the  springs  are  deduced  from  their  lengths 
and  calibration  curves. 

(2)  The  same  rod  with  its  attached  weight  is  held  at  30°  to  the 
horizontal  by  a  horizontal  force  at  the  lower  end,  and  an  inclined  force 
at  the  upper  end.  Find  the  magnitudes  of  the  forces  and  the  inclina- 
tion of  the  second  to  the  horizontal. 

This  problem  should  also  be  solved  both  theoretically  and 
experimentally. 

DISCUSSION 

(a)  Where  would  a  vertical  line  that  passes  through  the  intersec- 
tion of  the  lines  of  action  of  the  springs  intersect  the  rod?  (This 
might  form  part  of  the  exercise  and  be  tested  experimentally.) 

(5)  Solve  the  problems  (1)  and  (2)  by  a  method  indicated  by  the 
answer  to  (a). 

(c)  A  uniform  beam  rests  on  a  smooth  horizontal  rail  and  one  end 
of  it  presses  against  a  smooth  vertical  wall.  In  what  position  will  it 
be  in  equilibrium  ? 

(c?)  A  rod  hangs  from  a  hinge  on  a  vertical  wall  and  rests  on  a 
smooth  floor.  Calculate  the  pressure  on  the  floor  and  the  force  on  the 
hinge  if  the  mass  of  the  rod  be  1  kg. 

'(e)  A  uniform  ladder  30  ft.  long  rests  with  the  upper  end  against 
a  smooth  vertical  wall,  and  the  lower  end  is  prevented  from  slipping 
by  a  peg.  If  the  inclination  to  the  horizontal  be  30°,  find  the  pressure 
on  the  wall  and  at  the  peg,  the  ladder  weighing  100  Ib. 

(/")  A  uniform  rod  is  supported  by  means  of  two  strings  which  are 
attached  to  a  fixed  point  and  to  the  ends  of  the  rod.  Show  that  the 
tensions  of  the  strings  are  proportional  to  their  lengths. 

(•7)  Three  forces  acting  at  the  corners  of  a  triangle  each  perpen- 
dicular to  the  opposite  side  keep  the  triangle  in  equilibrium.  Show 
that  each  force  is  proportional  to  the  side  to  which  it  is  perpendicular. 


CHAPTER  VIII 

WORK  AND  ENERGY 

96.  Work.  —  The  scientific  conception  of  work  is  drawn 
from  many  of  the  most  common  experiences.  One  of  the 
oldest  of  these  is  the  effort  put  forth  in  raising  a  heavy 
body  to  a  higher  level,  e.g.  drawing  water  from  a  well  in  a 
bucket  or  carrying  stone  or  brick  up  a  ladder  or  stair  in 
building  a  house.  The  work  done  in  such  cases  evidently 
depends  on  at  least  two  things,  —  the  continued  effort  re- 
quired to  sustain  the  weight  of  the  body  and  the  height  to 
which  the  body  is  carried.  In  some  cases  the  work  done 
seems  to  depend  on  other  circumstances.  For  instance,  a 
man  can  raise  a  quantity  of  brick  to  a  certain  height  with 
less  expenditure  of  work  when  he  iises  a  pulley  than  when 
he  actually  carries  them  up.  But  in  the  latter  case  he 
carries  the  weight  of  his  body  up  also  and  the  work  of 
carrying  his  body  up  is  added  to  the  work  of  carrying  the 
brick  up.  Again,  more  work  is  required  to  draw  a  body 
up  a  rough  plane  to  a  certain  height  than  up  a  smooth 
plane  to  the  same  height ;  but  when  the  plane  is  rough 
the  force  of  friction  has  to  be  overcome  also  and  the  work 
done  against  friction  is  added  to  the  work  done  against 
gravity. 

A  consideration  of  such  cases  will  show  the  reasonable- 
ness of  measuring  the  work  done  in  moving  a  body  by  the 

tfiS 


134  DYNAMICS 

product  of  the  force  applied  to  the  body  and  the  distance 
through  which  it  is  applied. 

97.  Work  and  Direction  of  Motion.  —  By  the  distance 
through  which  a  force  is  applied  is  meant  the  distance 
measured  in  the  direction  of  the  force  applied  to  do  the 
work.  When  the  force  overcome  is  gravity  the  distance 
must  be  measured  vertically.  A  very  heavy  body  can 
under  suitable  circumstances,  e.g.  when  attached  to  a 
crane,  be  moved  horizontally  with  only  a  very  slight  ex- 
penditure of  work ;  and  the  slight  amount  expended  is 
due  to  the  fact  that  some  friction  has  to  be  overcome.  If 
there  is  no  vertical  motion,  the  work  done  against  gravity 
is  zero  whatever  work  may  be  done  against  other  forces, 
j  A  C  Hence  if  a  force  F  be 

exerted    a    distance    AB 
not  in  the  direction  of  the 
force,   the   work    done  is 
Fm- 53'  B  F .  AB  cos  0,  6  being  the 

angle  between  the  positive  direction  of  F  and  the  positive 
direction  of  AB.  But 

F'ABGosQ  =  Fcos0>  AB 

and  F  cos  0  is  the  component  of  F  in  the  direction  of  AB. 
Hence  the  work  done  by  a  force  may  also  be  measured  by 
the  product  of  the  displacement  and  the  component  of  the 
foree  in  the  direction  of  the  displacement. 

If  0  =  90°,  i.e.  if  the  displacement  is  altogether  at  right 
angles  to  the  force,  the  force  does  no  work.  Hence  the 
force  towards  the  centre  in  circular  motion  does  no 
work. 


WORK  AND  ENERGY  135 

98.  Units  and  Dimensions  of  Work.  —  The  unit  of  work  is 
the  work  done  by  unit  force  when  its  point  of  application 
moves  unit  distance  in  the  direction  of  the  force.  Hence 
in  the  absolute  C.G.S.  system  the  unit  of  work  is  the  work 
done  by  a  dyne  when  it  acts  through  a  centimetre.  This 
unit  is  called  the  erg  ;  10,000,000  ergs  is  called  a  joule. 
In  the  F.  P.  S.  gravitational  system  the  unit  of  work  is 
the  work  done  by  a  force  equal  to  the  weight  of  a 
pound  when  it  acts  through  a  foot  and  is  called  a  foot- 
pound. 

Since  W=  F  •  s,  the  dimensions  of  work  are 

(F)  = 

Hence  in  the  absolute  system 

(W)  = 


99.  Rate  of  doing  Work,  or  Activity.  —  The  work  done  by 
a  force  depends  only  on  the  magnitude  of  the  force  and 
the  extent  of  the  displacement,  and  is  independent  of  the 
time  required  for  the  motion.  The  amount  of  work  an 
agent  does  in  a  certain  time,  or  the  rate  of  doing  work,  is 
a  different  thing  and  is  a  matter  of  great  importance. 
(The  wealth  a  man  has  is  the  same  whether  it  took  him  a 
year  or  twenty  years  to  accumulate  it.  The  rate  at  which 
he  can  gather  wealth  is  a  different  thing.) 

Rate  of  doing  work  is  called  activity.  The  unit  of 
activity  in  the  absolute  C.  G.  S.  system  is  the  activity  of 
an  agent  that  does  an  erg  per  second  ;  10,000,000  ergs 
per  second,  or  a  joule  per  second,  is  called  a  watt;,  1000 
watts  is  called  a  kilowatt.  The  F.  P.  S.  gravitational 
unit  of  activity  is  550  foot-pounds  per  second,  and 


136 


DYNAMICS 


is   called   a   horse-power.      One   horse-power  =  746  watts 
very  nearly. 

100.  Diagram  of  Work.  —  The  work  done  by  a  constant 
force  is  readily  calculated  from  the  force  and  the  displace- 
ment.    Work   done  by  a  variable   force,   e.g.  the  force 
exerted  by  the  piston  of  a  steam  engine,  can  be  conven- 
iently represented  by  the  following  graphical  method  : 

On  a  straight  line,  which  we  may  suppose  horizontal, 
lengths  are  laid  off  to  represent  the  displacements  in  the 
direction  of  the  force  in  intervals  so  short  that  the  force  may 

be  regarded  as  constant 
throughout  each  inter- 
val. A  vertical  line  or 
ordinate  is  drawn  from 
the  middle  of  each  dis- 
placement to  represent 
the  magnitude  of  the 
force  at  the  middle  of 
^  the  interval.  A  smooth 
curve  is  then  drawn 
through  the  upper  end  of  the  ordinates.  The  area  be- 
tween the  curve,  the  horizontal  line,  and  any  two  ordi- 
nates represents  the  work  done  in  the  intervening  time. 
For  the  work  done  in  one  of  the  short  displacements,  such 
as  d  in  the  figure,  is  the  product  of  the  force  by  the  dis- 
placement, and  is  therefore  represented  by  the  area  of  the 
narrow  trapezium  that  stands  on  d. 

101.  Energy.  —  A  body  capable  of  doing  work  is  said 
to  possess  energy,  or  energy  is  defined  as  capacity  for 


d 
FIG.  54. 


WOEK  AND  ENERGY  137 

work.  A  body  can,  by  descending,  draw  up  an- 
other body  attached  to  it  by  a  cord  that  passes  over  a 
pulley.  A  compressed  or  extended  spring  can  raise  a 
weight.  Water  at  an  elevation  can  do  work  in  descend- 
ing to  a  lower  level.  These  are  examples  of  a  body  or 
a  system  of  bodies  possessing  energy  because  of  some 
peculiarity  in  its  form  or  position,  or,  briefly,  because  of 
its  configuration. 

The  block  of  a  pile-driver  can,  when  in  motion,  drive  a 
pile  in  opposition  to  the  forces  of  cohesion  and  friction. 
A  fly  wheel  in  rotation  can  for  a  time  keep  a  pump  going 
and  raise  water.  Wind  can  propel  a  ship  against  the 
resistance  of  the  friction  and  inertia  of  the  water.  These 
are  examples  of  a  body  possessing  energy  because  of  its 
mass  and  speed. 

Thus  we  have  two  chief  forms  of  energy  —  energy  of 
motion,  also  called  kinetic  energy,  and  energy  of  configura- 
tion, also  called  potential  energy  (potential  energy  must 
not  be  regarded  as  merely  possible  kinetic  energy,  for 
potential  energy  can  do  work  without  being  first  trans- 
formed into  kinetic  energy). 

102.  Kinetic  Energy.  —  The  energy  a  body  possesses 
because  of  its  mass  and  speed  should  be  capable  of  being 
expressed  in  terms  of  these  quantities.  To  discover  how 
it  should  be  expressed,  let  us  find  how  much  work  a 
body  of  mass  m  and  speed  v  can  do  in  coming  to  rest. 
Suppose  that  the  force  it  exerts  on  another  body  is 
variable,  and  that  while  acting  through  a  short  distance 
s1  it  exerts  a  force  Fr  Then  the  work  it  does  is  F^sr 
By  Newton's  Third  Law,  the  force  opposing  its  motion  is 


138  DYNAMICS 


—  J7,,  and  therefore  its  acceleration  is  a  = -•     If   its 

i/n 
speed  at  the  end  of  sx  be  vv  by  §  24 

2  as-, 


_ 

m 
or  -F^i  =  J  wv2  —  1  w^2. 

For  successive  small  displacements  sv  *2,  *8,  •••,  sw,  we 


Fnsn  =  J  mvw_!2  -  \ 

where  u  is  its  final  speed. 

Adding  these  equations,  we  get 


The  left  side  is  the  total  work  done,  and  if  the  final 
speed  of  the  body,  M,  is  zero,  its  kinetic  energy  is  ex- 
hausted, and  the  whole  work  it  has  done  is  \  mv2.  Hence 
this  was  its  initial  kinetic  energy. 

In  a  similar  way  it  can  be  shown  that  when  a  force  is 
opposed  only  by  the  inertia  of  a  body,  the  work  done 
equals  the  increase  of  |  mv2  in  the  body. 

The  units  and  dimensions  of  kinetic  energy  are  the 
same  as  those  of  work  (§  98). 

103.  Potential  Energy.  —  Quantities  of  potential  energy 
are  measured  by  the  work  they  can  do.  Thus,  when  the 
distance  of  a  body  from  the  earth  is  increased,  its  poten- 
tial energy  (or  more  correctly  the  potential  energy  of  the 
mass  and  the  earth)  is  increased  by  the  amount  of  work 


WORK  AND  ENERGY  139 

it  can  do  in  returning  to  its  first  position.  The  increase 
of  potential  energy  of  a  stretched  spring  is  the  work  it 
can  do  in  contracting. 

We  can  also  express  the  potential  energy  of  the  system 
consisting  of  the  earth  and  a  body  in  terms  of  their  masses 
and  distance  apart.  The  increase  of  potential  energ 
when  a  body  of  mass  m  grams  is  raised  a  distance  h  centi- 
metres (h  being  small  compared  with  the  radius  of  the 
earth)  is  mgJi  in  ergs,  for  this  is  the  work  it  can  do  in 
returning.  The  potential  energy  of  a  stretched  spring 
can  be  expressed  in  terms  of  its  length  and  elastic  con- 
stants. But  there  is  no  one  general  way  of  expressing 
potential  energy  as  there  is  in  the  case  of  kinetic  energy. 
For  each  body  or  system  we  must  find  by  experiment 
how  much  work  it  can  do  in  changing  from  one  configura- 
tion to  another  and  then  measure  the  charge  of  potential 
energy  by  the  work  so  done.  If  the  initial  potential 
energy  be  V\  and  the  final  potential  energy  be  F2,  the 
decrease  is  Vl  —  Vv  If  the  force  exerted  by  the  body  or 
system  during  successive  small  displacements  sv  $2,  •••  be 
Fv  F^  "•  respectively,  the  total  work  done  is  ^Fs.  Hence 
when  a  body  or  system  does  work  at  the  expense  of  its 
potential  energy 

J-  O»/ 


The  units  and  dimensions  of  potential  energy  are  the 
same  as  those  of  work  (§  98). 

104.  Equivalence  of  Kinetic  and  Potential  Energy.  —  In  the 
performance  of  work  energy  is  expended  and  the  energy  so 
expended  is  by  definition  equal  to  the  work  performed. 
But  when  work  is  done  on  a  system,  the  energy  of  the 


140  DYNAMICS 

system  is  increased.  Confining  ourselves  for  the  present  to 
cases  in  which  no  work  is  done  against  friction,  or,  in  other 
words,  supposing  the  work  is  wholly  performed  in  produc- 
ing kinetic  or  potential  energy,  then  the  energy  produced 
is  equal  to  the  work  performed,  that  is,  equal  to  the  energy 
expended. 

The  statements  in  preceding  sections  will  suggest  to  the 
reader  various  ways  in  which  energy  can  be  transferred 
froni  one  body  or  system  to  another,  the  former  doing 
work  on  the  latter.  Moreover  the  energy  in  a  single  sys- 
tem may  be  transformed  from  one  form  of  energy  to  the 
other  form.  Thus  when  a  body  is  allowed  to  fall  towards 
the  earth,  the  potential  energy  of  the  body  and  the  earth 
is  decreased,  but  their  kinetic  energy  is  (if  we  neglect  fric- 
tion) increased  to  an  equal  amount.  In  this  case  the  only 
force  affecting  the  system  is  an  internal  force  between  the 
different  parts  of  the  system  and  the  work  done  ly  an 
internal  force  results  in  a  transformation  of  the  energy  of  the 
system  without  any  change  in  its  amount. 

In  making  these  statements  we  have  supposed  that  fric- 
tion may  be  neglected.  As  a  matter  of  fact,  friction  can- 
not in  any  case  (except  possibly  in  the  case  of  the  motion 
of  the  heavenly  bodies)  be  wholly  neglected.  We  shall 
consider  later  the  case  of  work  done  against  friction. 

Exercise  XXIII.    Energy  and  Work 

Apparatus. — A  pendulum  with  a  heavy  block  of  iron  as  a  bob  is 
drawn  aside  and  held  in  any  desired  position  by  a  cord  arranged  as 
illustrated  in  Fig.  55.  When  the  cord  is  released  the  pendulum  falls 
and  impinges  on  a  horizontal  rod  which  is  connected  to  the  framework 
by  two  horizontal  springs.  The  springs  are  stretched  by  the  impact  and 


WORK  AND  ENERGY 


141 


the  pendulum  is  brought  momentarily  to  rest  when  it  has  just  reached 
the  vertical.     The  maximum  stretch  of  the  springs  is  recorded  by  a 


FIG.  56. 

loop  of  thread  that  is  pushed  along  the  rod  by  a  collar  in  which 
the  rod  slides.  The  rod  should  be  as  light  as  possible  ;  in  fact,  it  is 
better  to  use  a  tube  of  aluminium  instead  of  a  rod.  The  knife-edge 


142  DYNAMICS 

on  which  the  pendulum  swings  is  carried  by  two  plates,  one  of  which  is 
adjustable  by  two  screws.  The  necessary  readings  will  be  facilitated 
if  a  scale  is  lightly  etched  on  the  rod,  but  this  is  not  indispensable. 

Adjustments.  —  The  screws  that  support  the  knife-edge  are  adjusted 
until  the  pendulum  swings  freely  without  any  side  motion.  The  light 
chains  that  attach  the  springs  to  the  framework  are  adjusted  on  hooks 
in  the  framework  until  the  springs  are  under  slight  tension  and  do  not 
vibrate  sidewise  when  the  rod  is  pushed  rapidly  by  the  hand.  The 
proper  height  from  which  to  release  the  pendulum  so  that  it  just  comes 
to  rest  in  the  vertical  must  be  ascertained  by  several  trials.  To  fix  the 
vertical  position  of  the  pendulum,  the  rod  is  held  by  the  hand  with  the 
springs  stretched  so  that  its  end  just  presses  against  the  pendulum 
when  vertical;  the  thread  is  then  pushed  up  against  the  collar  and  its 
position  noted  (or  marked  in  pencil  on  the  rod  if  there  is  no  scale  on 
the  rod).  This  is  the  position  to  which  the  thread  must  be  moved  by 
the  impact  if  the  pendulum  comes  to  rest  in  the  vertical.  Before 
readings  are  made  the  rod  should  be  lubricated  with  vaseline  or 
machine  oil. 

Measurements.  —  (1)  The  length  of  the  pendulum  from  the  knife- 
edge  to  the  centre  of  the  bob.  (2)  The  length  of  the  cord  of  the  arc 
through  which  the  centre  of  gravity  of  the  pendulum  falls.  (3)  The 
initial  length  of  the  springs  and  the  extreme  length  to  which  they  are 
stretched ;  the  latter  is  obtained  from  the  movement  of  the  thread. 
(4)  Calibration  of  the  springs.  For  this  purpose  the  springs  may  be 
removed  and  calibrated  in  several  steps  through  the  range  covered  by 
the  experiment.  The  calibration  may  also  be  performed  without 
removing  the  springs,  by  attaching  to  the  rod  a  cord  that  passes  over  a 
pulley  and  carries  a  pan  and  weights. 

From  the  calibration  of  the  springs  a  diagram  of  work  done  by  the 
pendulum  in  stretching  the  springs  is  constructed  on  cross-section 
paper  and  the  work  calculated.  (The  scale  of  the  diagram  must  be 
taken  into  consideration  as  in  Exercise  IV.)  This  should  nearly 
equal  the  loss  of  potential  energy  of  the  pendulum  as  calculated 
from  its  mass  and  vertical  descent,  but  there  will  be  some  difference 
caused  by  friction  and  impact  (§  113). 


WORK  AND  ENERGY  143 

DISCUSSION 
(a)  Sources  of  error. 

(&)  Changes  that  take  place  in  the  energy  of  the  pendulum  and  of 
the  springs. 

(c)  Energy  of  rotation  of  the  bob  of  the  pendulum. 

(d)  The  total  energy  of  the  system. 

(e)  Are  the  results  affected  in  any  way  by  the  mass  of  the  rod  ? 
(/)  Why  must  the  initial  tension  of  the  springs  not  be  large? 
(</)   Explain  the  failure  of  the  pendulum  to  rise  after  rebound  to 

its  original  position. 

105.  Stable,  Unstable,  and  Neutral  Equilibrium. — A  body 
or  structure  is  in  equilibrium  when  the  resultant  force  on 
it  is  zero,  that  is,  when  it  is  either  at  rest  or  moving  uni- 
formly. A  body  or  structure  at  rest  is  in  stable  equilibrium 
when  on  being  displaced  it  returns  (e.g.  a  pendulum,  a 
sphere  in  a  bowl,  a  chemical  balance);  it  is  in  unstable 
equilibrium  when  on  being  displaced  it  moves  farther  away 
(e.g.  an  egg  on  one  end,  a  rigid  pendulum  inverted) ;  it  is 
in  neutral  equilibrium  when  on  being  displaced  it  remains 
at  rest  (e.g.  a  sphere  on  a  horizontal  plane,  a  body  that 
can  rotate  about  an  axis  through  its  centre  of  gravity). 

If  displacement  causes  an  increase  of  potential  energy, 
it  is  evident  that  work  has  been  done  against  forces  oppos- 
ing the  displacement,  and  these  forces  will  cause  the  body 
or  structure  to  return.  Hence,  a  position  of  stable  equi- 
librium is  a  position  of  minimum  potential  energy.  The 
centre  of  gravity  of  a  pendulum  or  a  balance  is  raised  by 
a  displacement  and  its  potential  energy  is  increased. 

If  the  displacement  produces  a  decrease  of  potential 
energy,  the  forces  acting  must  have  aided  the  displace- 
ment, and  they  will  therefore  still  further  increase  the 


144 


DYNAMICS 


displacement.  Hence,  a  position  of  unstable  equilibrium  is 
a  position  of  maximum  potential  energy.  If  an  egg  be  sup- 
posed balanced  on  one  end,  its  centre  of  gravity  will  be 
lowered  by  a  displacement,  and  therefore  the  potential 
energy  will  be  decreased. 

Finally,  if  when  a  body  or  structure  is  displaced  there 
is  no  change  of  potential  energy,  then  the  forces  acting 
on  the  body  neither  oppose  nor  assist  the  motion,  and 
hence  the  equilibrium  will  be  neutral. 


FIG.  56. 

The  energy  criterion  of  equilibrium  may  be  illustrated 
by  the  apparatus  represented  in  the  diagram.  A  light 
jointed  parallelogram  is  attached  to  an  upright  post  by 
horizontal  axes  passing  through  the  centres  of  two  oppo- 
site sides.  Equal  heavy  weights  are  movable  along  rods 
fastened  at  right  angles  to  the  vertical  sides.  The  appa- 
ratus is  in  neutral  equilibrium  no  matter  where  the  weights 
may  be  on  the  rods,  for  no  work  is  done,  and  there  is  no 
change  of  potential  energy  during  a  displacement. 

106.  Kinetic  Energy  of  a  Rotating  Body. — When  a  body 
rotates  about  a  fixed  axis  with  angular  velocity  o>,  a  par- 


WORK  AND  ENERGY  145 

tide,  w,  at  a  distance,  r,  from  the  axis  has  a  linear  velocity 
G>r,  and  therefore  its  kinetic  energy  is  J  mco2r2.  The  ki- 
netic energy  of  the  whole  body  is  therefore  E  = 
But  since  o>  is  the  same  for  all  the  particles, 


Z  being  the  moment  of  inertia  of  the  body  about  the  axis 
of  rotation.  The  similarity  of  this  formula  to  the  formula, 
\  mv2,  for  the  kinetic  energy  of  translation  of  a  body 
should  be  noted. 


Exercise  XXIV.    Kinetic  and  Potential  Energy 

In  Exercise  XVII  a  mass  descended  losing  potential  energy,  and 
in  so  doing  it  set  into  rotation  a  disk  which  acquired  kinetic  energy. 
The  descending  mass  also  gained  some  kinetic  energy.  The  circum- 
stances were  such  that  the  resistance  of  friction  was  very  small. 
Hence,  the  gain  of  kinetic  energy  should  be  (at  least  very  nearly) 
equal  to  the  loss  of  potential  energy.  These  quantities  can  be  calcu- 
lated from  the  observations  then  made.  It  may,  however,  be  well  to 
repeat  the  measurements  of  distance  and  time  of  descent  with  all  the 
care  possible. 

To  calculate  the  kinetic  energy  the  final  angular  velocity  must  be 
known.  This  can  be  deduced  from  the  time  of  descent  and  the  dis- 
tance of  descent.  For  these  give  the  mean  velocity  of  descent,  and 
twice  this  is  the  final  velocity.  From  the  final  linear  velocity  of 
descent  and  the  radius  of  the  axis,  the  final  angular  velocity  of  the 
disk  is  deduced.  Thus  we  have  all  the  data  necessary  to  test  the 
equality  of  loss  and  gain  of  energy. 

(The  apparatus  of  Exercise  XVIII  may  also  be  used  for  this 
experiment,  the  final  velocity  of  rotation  being  obtained  directly  by 
means  of  the  recording  disk.  On  account  of  the  greater  air  friction 
the  results  will  probably  not  be  found  as  satisfactory  as  those 
obtained  by  the  method  described  above.) 


146  DYNAMICS 

DISCUSSION 

(a)  Sources  of  error. 

(b)  Is  the  friction  necessarily  assumed  to  be  zero  in  the  method  of 
finding  the  final  velocity  ? 

(c)  Find  an  expression  for  the  final  kinetic  energy  that  does  not 
contain  the  final  linear  velocity;  also  one  that  does  not  contain  the 
final  angular  velocity. 

(e?)  What  would  the  final  angular  velocity  be  if  the  thread  were 
wrapped  around  the  disk  ? 

(e)  Given  the  coefficient  of  friction  between  the  cylinders  and  the 
disk,  at  what  angular  speed  would  the  former  slide  if  not  restrained 
by  pegs  ? 

(_/*)  If  the  friction  between  the  cylinders  and  the  disk  prevented 
slipping,  at  what  angular  speed  would  the  cylinders  be  overturned? 

(g)  What  form  of  cylinders  would  render  slipping  and  overturning 
equally  probable  ? 

(A)  Method  of  measuring  the  coefficient  of  static  friction  suggested 
by  this  exercise. 

(i)  If  the  thread  remained  attached  to  the  axis  and  the  weight  just 
touched  the  floor,  how  much  would  the  angular  velocity  decrease  when 
the  thread  began  to  rewind  ? 

(/)  If  the  thread  remained  attached  to  the  axis  and  the  weight  did 
not  touch  the  floor,  how  much  would  the  angular  velocity  decrease 
when  the  cord  began  to  rewind  ? 

107.  Simultaneous  Translation  and  Rotation.  —  We  have 
already  seen  that  the  motion  of  a  body  may  be  considered 
as  consisting  of  velocity  of  translation  of  the  centre  of 
mass,  and  velocity  of  rotation  about  the  centre  of  mass, 
and  that  these  two  motions  may  be  regarded  as  inde- 
pendent, and  may  be  calculated  separately  (§  86).  Simi- 
larly the  whole  kinetic  energy  of  a  body  may  be  considered 
as  consisting  of  two  parts  corresponding  to  these  two 
motions.  If  M  is  the  whole  mass  of  the  body,  and  V 


WORK  AND  ENERGY  147 

the  velocity  of  the  centre  of  mass,  the  kinetic  energy 
of  translation  is  \MV^.  If  the  body  has  an  angular 
velocity,  o>,  about  an  axis  through  the  centre  of  mass, 
and  if  /  is  the  moment  of  inertia  about  that  axis,  the 
kinetic  energy  of  rotation  is  \I^>  The  total  energy 
is  the  sum  of  these  two.  Thus  the  kinetic  energy  of  a 
locomotive  wheel  can  be  calculated  when  the  velocity 
of  its  centre,  and  its  angular  velocity  about  its  centre, 
are  known. 

The  same  principle  applies  to  the  motion  of  any  group 
of  particles,  but,  if  the  particles  are  not  connected  rigidly 
together,  they  have  no  angular  velocity  in  common,  and 
their  kinetic  energy,  relatively  to  the  centre  of  mass,  must 
be  found  in  a  different  way.  Suppose  that  any  particle, 
m,  has,  relatively  to  the  centre  of  mass  of  all  the  particles, 
a  velocity  V ;  its  kinetic  energy  of  motion,  relatively  to 
the  centre  of  mass,  is  \mV<2t.  Summing  up  for  all  the 
particles,  we  get  the  expression  SJmJ72  for  the  kinetic 
energy  of  all,  relatively  to  their  centre  of  mass.  This, 
together  with  \MV\  makes  up  the  whole  kinetic  energy 
of  the  group  of  particles. 

Since  internal  forces  do  not  affect  the  motion  of  the 
centre  of  mass  of  a  body  or  group  of  particles  (§  85),  they 
cannot  change  the  part  of  the  kinetic  energy  that  depends 
on  the  motion  of  the  centre  of  mass. 

The  propriety  of  dividing  kinetic  energy  into  these  two  parts  needs, 
in  reality,  somewhat  more  proof  than  has  been  given.  Suppose  the 
velocity  V  of  the  centre  of  mass  to  be  resolved  into  rectangular  com- 
ponents u,  v,  w.  Also  let  F,  the  velocity  of  any  particle,  w,  relatively 
to  the  centre  of  mass,  be  resolved  into  components,  u,  v,  w.  Then  the 
whole  velocity,  relatively  to  the  origin,  of  the  particle,  m,  has  com- 
ponents u  +  w,  v  +  v,  w  4-  w,  and  its  square  is  therefore  equal  to  the 


148  DYNAMICS 

sum  of  the  squares  of  these  components.     Hence  the  whole  kinetic 
energy  is 


2+  (o  +  v)2 
w2  +  y2  +  w2)  +  2  1  m  (w2  +  v2  + 


The  last  three  terms  are  all  zero  since  2,mu  =  'Zmv  =  ^?nw  =  0  (§  84). 
The  first  term  is  readily  seen  to  be  |  MV2,  since  w,  y^  w,  are  the  com- 
ponents of  V.  The  second  term  is  5jmF2,  and  in  the  case  of  a  rigid 
body  this  is  equal  to  1  7o>2  (§  106). 

108.  Work  done  by  the  Moment  of  a  Force  or  Couple.  — 
A  force,  F,  acting  at  a  point,  P,  of  a  body  free  to  rotate 
about  an  axis,  A,  will  produce  a  displacement  of  P  and  will 

therefore  do  work.  Let 
/  be  the  component  of  F 
in  the  direction  of  motion 
of  P,  and  let  r  be  the  dis- 
tance of  P  from  the  axis 
A.  If  F  be  of  constant 
magnitude  and  if  its  direction  with  reference  to  AP  be 
also  maintained  constant,  then  /  will  be  of  constant 
magnitude.  When  the  body  turns  through  an  angle  0, 
P  will  move  through  a  distance  rO,  and  the  work  done 
by  F  will  be  frd.  But  since  /  is  the  only  component 
of  F  that  has  a  moment  about  A,  the  moment  of  F  about 
A  is  fr.  Denoting  it  by  (7,  the  work  done  by  0  in  an 
angular  displacement  6  will  be  CO.  If  the  force  be 
entirely  exerted  in  producing  kinetic  energy  of  rotation 
about  A,  then 


a)  being  the  initial  angular  velocity,  and  o^  the  final  angu- 
lar velocity. 


WOEK  AND  ENERGY  149 

If  F  be  not  constant  or  if  its  direction  with  reference  to 
AP  vary,  then/  and  also  C  will  be  variable.  In  this  case 
the  work  done  will  be  2(70,  (7  being  the  moment  of  the 
force  during  a  small  displacement  6. 

Exercise  XXV.     Kinetic  and  Potential  Energy 

In  Exercise  IX  no  account  was  taken  of  the  mass  of  the  wheel 
nor  of  the  force  required  to  overcome  the  friction  of  the  bearings  of 
the  wheel.  It  is  now  proposed  to  take  account  of  these  and  treat  the 
problem  from  the  point  of  view  of  gain  and  loss  of  energy.  The 
observations  described  under  "distance  and  acceleration"  in  Ex- 
ercise IX  may  be  repeated;  or,  if  exactly  the  same  apparatus  is 
used,  the  results  obtained  in  that  exercise  may  be  used;  but  the 
former  is  preferable,  since  doubt  would  remain  as  to  whether  the 
apparatus  was  in  exactly  the  same  condition  in  all  respects. 

Loss  of  Potential  Energy. —  The  resultant  loss  of  potential  energy 
is  readily  calculated  from  the  loss  of  potential  energy  of  the  total 
descending  mass  and  the  gain  of  potential  energy  of  the  ascending 
mass.  The  items  should  be  stated  in  ergs  or  joules. 

Kinetic  Energy  of  Masses.  —  From  the  acceleration  and  distance  the 
final  velocity  of  the  masses  can  be  calculated  and  thence  the  kinetic 
energy  deduced. 

Friction.  —  The  friction  of  the  bearings  of  the  wheel  varies  some- 
what with  the  position  of  the  wheel.  It  may  be  measured  though  not 
very  accurately  by  finding  what  additional  weight  placed  on  one  of 
the  masses  will  just  keep  the  (equal)  masses  in  steady  motion  when 
once  started.  This  should  be  repeated  for  several  different  positions 
of  the  cord  on  the  wheel  and  the  mean  taken.  The  work  done  against 
friction  equals  the  product  of  the  moment  of  the  force  that  will  over- 
come the  friction  and  the  total  angle  through  which  the  wheel  turns. 

Kinetic  Energy  of  Wheel.  —  The  kinetic  energy  gained  by  the  wheel 
can  be  calculated  from  its  moment  of  inertia  and  final  angular  velocity. 
The  final  angular  velocity  of  the  wheel  is  readily  deduced  from  the 
relation  between  the  angular  velocity  of  the  wheel,  the  linear  velocity 


150  DYNAMICS 

of  a  point  on  its  circumference  (which  is  the  same  as  the  linear  veloc- 
ity of  the  cord),  and  the  radius  of  the  groove  on  the  wheel. 

The  moment  of  inertia  of  the  wheel  may  be  found  by  observing 
the  angular  acceleration  that  a  known  moment  of  force  will  give  to 
the  wheel.  Remove  the  large  masses  and  the  wire  and  wrap  a  light 
cord  around  the  wheel,  fastening  one  end  to  a  spoke.  To  the  free  end 
attach  a  weight  and  find  the  height  to  which  the  weight  must  be 
raised  so  that  when  the  wheel  is  released  on  a  tick  of  the  clock,  the 
weight  will  just  strike  the  floor  after  some  exact  number  of  seconds. 
To  allow  for  friction,  find  as  before  what  small  weight  attached  to 
the  cord  will  just  keep  the  wheel  in  steady  motion. 

109.  Conservative  and  Dissipative  Forces.  —  In  stating 
the  equivalence  of  kinetic  and  potential  energy  during 
transference  or  transformation,  we  limited  the  statement 
to  cases  in  which  energy  is  expended  while  exerting  forces 
that  are  wholly  employed  in  producing  kinetic  or  poten- 
tial energy.  In  such  cases  the  total  quantity  of  kinetic 
and  potential  energy  is  unchanged,  or,  as  it  is  frequently 
stated,  conserved.  The  forces  through  whose  agency  such 
transference  and  transformation  are  effected  are  called 
conservative  forces.  Examples  are  the  force  of  gravita- 
tion and  the  force  exerted  by  a  compressed  spring. 

If  we  examine  such  conservative  forces,  we  shall  find 
that  they  have  one  characteristic  in  common.  They  are 
known  when  the  positions  or  configurations  of  the  bodies 
are  assigned.  The  force  of  gravitation  on  a  body  at  a 
certain  height  is  the  same  whether  the  body  be  at  rest  or 
moving  in  any  way.  The  force  exerted  by  a  spring 
depends  only  on  its  length  at  a  certain  moment  and  not 
on  whether  it  is  contracting  or  expanding. 

Why  the  energy  should  remain  constant  when  the  only 
forces  acting  are  conservative  forces  can  be  readily  seen. 


WORK  AND  ENERGY  151 

Consider  a  body  started  upward  along  a  smooth  plane. 
While  rising  it  is  acted  on  at  each  point  by  a  force  of  a 
certain  magnitude,  and  so  does  work  and  therefore  loses 
kinetic  energy  to  an  extent  depending  only  on  the  force 
at  each  point  and  the  displacement.  But  exactly  the 
same  force  will  act  on  it  at  each  point  when  it  descends, 
and  therefore  while  rising  it  is  gaining  power  of  doing 
work  or  potential  energy  equal  to  the  work  done  or 
kinetic  energy  lost  in  rising.  Hence  its  total  energy  re- 
mains constant. 

But  now  suppose  the  same  body  started  up  a  rough 
plane.  Then  a  second  force,  friction,  acts  on  it.  This 
opposes  its  rise  and  will  also  oppose  its  descent.  Hence 
the  resultant  force  on  the  body  will  be  less  at  any  height 
during  the  descent  than  it  was  during  the  rise.  There- 
fore, in  rising  it  does  not  accumulate  power  of  doing 
work  in  the  form  of  potential  energy  equal  to  the  loss  of 
kinetic  energy. 

Friction  is  thus  a  non-conservative,  or,  as  it  is  fre- 
quently called,  a  dissipative  force.  It  depends  on  the 
direction  (and  often  on  the  magnitude)  of  the  velocity  of 
the  body  on  which  it  acts.  Other  dissipative  forces  more 
or  less  analogous  to  friction  will  be  met  with  in  the 
special  branches  of  physics. 

For  an  obvious  reason  conservative  forces  are  sometimes 
called  positional  forces,  and  dissipative  forces  are  some- 
times called  motional  forces. 

110.  The  Conservation  of  Energy.  —  A  system  of  bodies 
that  neither  does  work  on  outside  bodies  nor  has  work 
done  on  it  by  outside  bodies  is  an  isolated  system.  While 


152  DYNAMICS 

no  system  is  completely  isolated,  many  are  nearly  so,  and 
may  for  most  purposes  be  regarded  as  isolated.  The 
whole  solar  system  is  perhaps  the  best  example.  Except 
for  the  atmosphere,  the  earth  and  a  projectile  might  be 
treated  as  an  isolated  system.  The  earth,  a  projectile, 
and  the  atmosphere  is  a  still  closer  approach  to  an  isolated 
system.  A  spring  or  tuning-fork  vibrating  in  a  vacuum 
would  be  almost  an  isolated  system. 

When  an  isolated  system  does  work  against  internal 
dissipative  forces,  the  amount  of  work  so  done  represents 
an  equal  amount  of  kinetic  or  potential  energy  subtracted 
from  the  system.  Hence,  if  to  the  sum  of  the  kinetic  and 
potential  energies  of  an  isolated  system  we  add  the  work 
done  against  dissipative  forces,  the  whole  sum  is  constant. 

What  becomes  of  the  energy  spent  in  doing  work 
against  dissipative  forces  ?  It  was  long  supposed  to  be 
wholly  annihilated.  Newton  was  aware  of  the  constancy 
of  the  sum  of  what  we  now  call  kinetic  and  potential 
energy  and  work  done  against  dissipative  forces,  but  it 
was  nearly  two  centuries  before  it  was  recognized  that  the 
work  done  against  dissipative  forces  gives  rises  to  a  store 
of  energy  equal  to  that  expended  in  doing  the  work.  It 
was  then  found  that  such  work  produces  an  amount  of 
heat,  that,  measured  in  heat  units,  is  in  all  cases  exactly 
proportional  to  the  energy  expended,  or,  in  other  words, 
that  heat  is  equivalent  to  energy  and  is  therefore  itself  a 
form  of  energy.  Previous  to  that  time,  heat  was  sup- 
posed to  be  a  very  light  form  of  matter  called  caloric. 

The  new  view  naturally  suggested  that,  if  we  could  fol- 
low the  motion  of  the  particles  of  a  body  that  becomes 
heated,  we  would  find  that  each  particle  has  a  store  of 


WORK  AND  ENERGY  153 

kinetic  and  potential  energy,  and  that,  moreover,  in  the 
changes  from  one  form  to  the  other  and  from  particle  to 
particle  only  conservative  forces  come  into  play.  When 
a  body  slides  down  a  rough  plane,  the  change  of  some  of 
the  energy  of  motion  of  the  whole  body  into  energy  of 
motion  of  the  separate  particles  of  the  body  and  of  the 
plane  may  be  effected  by  conservative  forces  between  the 
particles.  From  this  point  of  view  the  distinction  be- 
tween conservative  and  dissipative  forces  would  disappear. 
But  the  belief  that  heat  is  a  form  of  energy  is  not 
founded  on  any  view  as  to  the  state  of  the  particles  of  a 
body  that  is  heated.  The  belief  is  founded  on  the  fact 
that  heat  and  other  forms  of  energy  are  interchangeable 
and  numerically  equivalent.  Including  heat  as  a  form  of 
energy  and  also  other  forms  of  energy  that  will  be  treated 
in  the  special  branches  of  physics  we  may  state  the  law  of 
the  Conservation  of  Energy  thus :  "  The  total  energy  of 
any  material  system  is  a  quantity  which  can  neither  be 
increased  nor  diminished  by  any  action  between  the  parts 
of  the  system,  though  it  may  be  transformed  into  any  of 
the  forms  of  which  energy  is  susceptible  "  (Maxwell). 

111.  The  Dissipation  of  Energy.  —  We  have  seen  that 
from  one  point  of  view  the  distinction  between  conserva- 
tive and  dissipative  forces  seems  to  disappear.  But  from 
another  point  of  view  there  seems  a  more  permanent  dis- 
tinction between  them.  Work  done  against  conservative 
forces  produces  forms  of  energy  that  can  be  confined  to 
definite  portions  of  matter.  For  example,  the  potential 
energy  of  the  earth  and  a  heavy  body  seems  to  remain 
definitely  associated  with  them,  and  the  potential  energy 


154  DYNAMICS 

of  a  distorted  spring  seems  to  have  no  tendency  to  escape. 
On  the  other  hand,  work  done  against  dissipative  forces 
gives  rise  to  forms  of  energy  that  tend  to  diffuse  without 
limit.  The  heat  produced  by  work  against  friction 
spreads  from  molecule  to  molecule  of  the  bodies  in  con- 
tact and  from  them  to  other  adjacent  bodies,.  This  diffu- 
sion of  energy  wherever  dissipative  forces  are  in  action  is 
called  the  dissipation  of  energy.  It  is  a  process  always 
going  on,  for  dissipative  forces  are  present  wherever 
changes  of  any  kind  are  taking  place  in  Nature. 

The  principle  of  the  dissipation  of  energy  must  not  be 
regarded  as  at  variance  with  the  law  of  the  conservation 
of  energy.  The  former  refers  merely  to  the  distribution 
of  energy,  the  latter  to  the  constancy  of  the  quantity  of 
energy. 

112.  Impact.  —  When  two  spheres  moving  in  the  line 
joining  their  centres  impinge,  there  are  two  stages  to  the 
impact :  (1)  they  compress  each  other  until  they  come 
relatively  to  rest;  (2)  they  then  recover  partially  or 
wholly  and  push  one  another  apart.  During  both  stages 
they  repel  one  another  with  forces  that  are  by  Newton's 
Third  Law  equal  and  opposite.  Hence  they  suffer  equal 
and  opposite  changes  of  momentum,  or  the  total  momen- 
tum is  unchanged  by  the  impact. 

A  simple  relation  between  the  relative  velocities  of  the 
spheres  before  and  after  impact  was  discovered  experi- 
mentally by  Newton.  Let  us  suppose  that  the  spheres 
are  of  the  same  material,  are  homogeneous,  that  is,  have 
the  same  properties  at  all  parts  of  their  mass,  and  are 
isotropic  or  have  at  any  point  the  same  properties  in  all 


WOEK  AND  ENERGY  155 

directions.  Then  the  ratio  of  the  velocity  of  separation 
after  impact  to  the  velocity  of  approach  before  impact  is 
constant,  that  is,  is  independent  of  the  sizes  of  the  spheres 
and  their  separate  velocities,  and  depends  only  on  the 
material  of  which  they  consist.  This  law  has  been  shown 
by  others  to  hold  true  only  within  certain  limits  of  velocity. 
The  ratio  of  the  velocity  of  separation  to  the  velocity  of 
approach  is  called  the  "  coefficient  of  restitution  "  of  the 
material. 

If  the  spheres  are  not  homogeneous,  the  coefficient  of 
restitution  depends  on  the  properties  of  the  spheres,  at 
the  points  of  contact.  If  they  are  not  isotropic  (e.g.  wood), 
the  coefficient  depends  on  the  direction  of  the  grain  rela- 
tively to  the  line  of  impact. 

Let  the  masses  of  the  spheres  be  m^  and  mv  their  respec- 
tive velocities  before  impact  u^  and  u^  and  after  impact  v1 
and  v2  ;  then  from  the  constancy  of  the  total  momentum 

we  have 

(1) 


Consider  the  case  in  which  m^  is  ahead  before  impact 
and  both  m1  and  w2  are  moving  in  the  positive  direction. 
The  velocity  of  approach  is  w2  —  uv  and  of  separation, 
vl  —  v2.  Hence,  by  definition  of  the  coefficient  of  resti- 
tution, 


or  vl-v^  =  -e(ul-u^).  (2) 

From  (1)  and  (2),  v1  and  v2  can  be  calculated.  Care 
must  be  taken  to  give  the  numerical  values  of  wx  and  u% 
their  proper  signs, 


156  DYNAMICS 

If  the  impact  -of  two  smooth  spheres  be  oblique,  that  is, 
if  the  spheres  be  not  moving  before  impact  in  the  line  of 
their  centres,  then,  since  the  pressure  between  the  spheres 
is  altogether  in  the  line  joining  their  centres,  only  the 
components  of  their  velocities  along  that  line  will  be 
affected,  and  the  above  equations  will  apply  to  those  com- 
ponents only.  The  components  perpendicular  to  the  line 
of  the  centres  will  be  unchanged. 

113.  Dissipation  of  Energy  on  Impact.  —  The  forces  that 
come  into  play  during  impact  are  not  wholly  conservative. 
If  the  material  of  the  impinging  bodies  is  plastic,  as  in 
the  case  of  putty  or  lead,  the  forces  tending  to  produce 
separation  are  small,  and  e  is  small.  The  kinetic  energy 
of  the  system  is  partly  spent  in  deforming  the  materials, 
doing  work  against  cohesion  and  internal  friction.  Even 
if  the  materials  recover  wholly  from  deformation,  still, 
during  the  deformation  and  recovery,  work  is  done  against 
internal  friction,  and  heat  and  sound  are  produced.  Thus 
energy  is  dissipated  and  the  kinetic  energy  after  impact 
is  less  than  that  before.  The  amount  so  dissipated  can 
be  found  from  the  masses  and  their  velocities. 

Exercise  XXVI.    Impact 

Apparatus.  —  A  ball  of  ivory  or  wood  forming  the  bob  of  a  pendu- 
lum impinges  on  another  ball  suspended  similarly  and  initially  at 
rest.  Each  supporting  cord  is  in  the  form  of  a  V,  in  order  that  the 
motion  of  the  ball  may  be  confined  to  a  vertical  plane.  The  imping- 
ing ball  should  be  dropped  from  some  definite  position  without  any 
jar  at  starting.  To  facilitate  this,  the  apparatus  is  provided  with  a 
rod,  in  the  end  of  which  a  needle  is  thrust ;  over  the  end  of  the  needle 
a  small  loop  of  thread  attached  to  the  ball  is  passed.  When  the  ball 
has  come  to  rest,  the  loop  of  thread  is  gently  pushed  off  the  needle, 


WORK  AND  ENERGY 


157 


and  so  the  ball  is  released  with  very  little  jar.     A  block  carrying  two 
vertical  wires  (or  knitting  needles)  and  movable  along  a  horizontal 


FIG.  58. 

metre  stick  is  used  to  measure  the  horizontal  distances  through  which 
each  ball  moves.     The  wires  should  be  made  vertical  by  a  square. 

Adjustments.  —  The  supporting  cords  are  adjusted  until  the  imping- 
ing ball  is  moving,  at  the  moment  of  impact,  in  the  line  joining  the 


158  DYNAMICS 

centres  of  the  two  balls  and  the  second  ball  moves  off  in  the  same  line. 
The  distance  between  the  upper  ends  of  the  cords  must  be  made  equal 
to  the  sum  of  the  radii  of  the  balls  so  that  when  the  pendulums  are 
at  rest  the  balls  just  touch  without  pressing  against  one  another. 

Observations.  —  The  velocity  of  each  ball  before  impact  and  also  its 
velocity  after  impact  are  to  be  found.  The  velocity  can  be  deduced  if 
the  height  from  which  each  ball  falls  or  the  height  to  which  it  rises  is 
known.  The  height  is  too  small  to  be  measured  directly  with  any 
degree  of  accuracy;  but  the  horizontal  distance  traversed  can  be 
measured  with  considerable  accuracy  by  means  of  the  movable  block 
and  the  horizontal  scale ;  and,  from  the  horizontal  distance  and  the 
radius  of  the  circle  of  motion,  the  height  can  be  deduced. 

In  the  case  of  the  impinging  ball  the  measurement  of  the  horizon- 
tal distance  is  made  by  adjusting  the  movable  block  until  the  ball  is 
tangential  to  the  plane  of  the  upright  wires,  (1)  when  the  ball  is  in 
its  elevated  position,  arnd  (2)  when  the  ball  is  at  rest  in  its  lowest 
position,  and  reading  the  position  of  the  block  on  the  scale  in  each 
position.  To  obtain  the  height  to  which  each  ball  rises  after  impact 
the  movable  block  is  adjusted  until  the  ball  as  it  rises  just  seems  to 
touch  the  plane  of  the  wires.  Several  trials  will  be  needed  in  each 
case  to  accomplish  this  adjustment. 

In  this  way  the  velocity  of  the  impinging  ball  before  impact  and 
the  velocity  of  each  ball  after  impact  are  measured.  From  these 
velocities  and  the  masses  of  the  balls  (which  can  be  obtained  by 
weighing)  the  total  momentum  and  the  total  kinetic  energy  before 
impact  and  the  total  momentum  and  total  kinetic  energy  after  impact 
are  calculated.  The  coefficient  of  restitution  should  also  be  calculated. 

The  experiment  should  be  performed  both  with  balls  of  equal 
masses  and  with  balls  of  unequal  masses. 

DISCUSSION 

(a)  Meaning  and  derivation  of  formulae. 

(&)   Sources  of  experimental  error. 

(c)  Deduction  of  the  ratio  of  the  masses  of  two  bodies  from  the 
changes  of  velocity  when  one  impinges  on  the  other.  Distinction 
between  mass  and  weight. 


WORK  AND  ENERGY  159 

(e?)  What  becomes  of  the  energy  lost  on  impact  ? 

(e)  How  is  the  motion  of  the  centre  of  gravity  of  the  two  balls 
affected  by  the  impact  ? 

(/)  A  50-gm.  bullet  is  fired  into  a  ballistic  pendulum  whose  mass 
is  50  kg.  If  the  velocity  of  the  bullet  is  300  m.  per  second,  what  'is 
the  velocity  and  momentum  of  the  pendulum? 

(<?)  Find  the  loss  of  kinetic  energy  in  (/"). 

(A)  A  ball  falls  from  a  height  of  16  ft.  and  rebounds  from  a  stone 
slab  to  a  height  of  8  ft.  Find  the  coefficient  of  restitution. 

(t)  A  Maxim  gun  fires  5  bullets  per  second  each  of  mass  30  g.  and 
having  an  initial  velocity  of  500  m .  per  second.  What  force  is  neces- 
sary to  hold  the  gun  at  rest  ? 

(j)  To  what  extent  is  Exercise  XXIII  a  case  of  impact? 

114.  Dissipation  of  Energy  of  Rotation.  —  Exercise  XVIII, 
on  the  constancy  of  the  angular  momentum  of  a  system 
whose  moment  of  inertia  changes,  affords  an  illustration  of 
the  dissipation  of  energy  of  rotation.  If  the  moments  of 
inertia  before  and  after  the  change  be  1^  and  J2  respec- 
tively, and  the  angular  velocities  o^  and  o>2  respectively, 
then  the  corresponding  values  of  the  kinetic  energy  are 
\  I-1&1  and  J  Itf>>z>  Since  Ila)1  =  /2o>2,  the  ratio  of  the 
kinetic  energy  after  the  change  to  that  before  the  change 
is  ct>2  :  o>r 

The  energy  lost  is  expended  in  work  against  the  friction 
between  the  sliding  blocks  and  the  cross-arm  and  in  the 
production  of  heat  and  sound  on  the  impact  of  the  blocks 
on  the  stops.  To  draw  the  blocks  back  again  to  their 
original  positions,  thus  increasing  the  angular  velocity  and 
restoring  the  lost  kinetic  energy,  work  would  have  to  be 
done  equal  to  the  energy  dissipated.  To  accomplish  this  a 
force  would  have  to  be  applied  to  the  cord  equal,  at  each 
stage,  to  the  sum  of  the  centrifugal  force  and  friction  and 


160  DYNAMICS 

the  total  work  done  by  the  force  applied  would  equal  the. 
increase  of  kinetic  energy. 

If  during  the  outward  movement  of  the  blocks  they 
were  compelled  to  draw  up  a  weight  and  if  the  weight  were 
of  such  magnitude  that  the  blocks  just  reached  the  stops 
without  impinging,  the  total  kinetic  and  potential  energy 
of  the  system  including  the  weight  would  be  unchanged 
except  for  the  energy  expended  against  friction.  Or  the 
blocks  might  be  compelled  to  stretch  springs  attached  to 
the  vertical  axis,  and  then  the  potential  energy  of  the 
springs  would  (neglecting  friction)  be  equal  to  the  loss  of 
kinetic  energy. 

Exercise  XXVII.    Angular  Momentum  and  Kinetic  Energy  of 

Rotation 

The  change  of  kinetic  energy  that  accompanied  the  change  of  mo- 
ment of  inertia  in  Exercise  XVIII  can  be  calculated  from  the  initial 
and  the  final  moment  of  inertia  and  the  initial  and  the  final  angular 
velocity  as  indicated  above.  It  can  also  be  found  by  the  following 
experimental  method. 

Let  the  kinetic  energy  before  the  change  of  moment  of  inertia  be 
Er  Attach  the  thread  (see  Fig.  46)  to  the  axis  so  that  it  will  not  be 
detached  when  wholly  unwrapped  and  arrange  the  cord  that  restrains 
the  blocks  so  that  the  moment  of  inertia  shall  not  change.  Then  the 
thread  after  unwinding  will  be  rewound  on  the  axis  and  the  weight 
m  will  rise  finally  to  a  height  h.  This  will  be  less  than  the  height  H, 
from  which  m  originally  descended  owing  to  the  effects  of  friction. 
If  W  be  the  work  done  against  friction  during  the  ascent  of  m,  E^  — 
mgh  +  W.  If  the  experiment  be  repeated,  the  cord  having  been  ar- 
ranged so  that  the  moment  of  inertia  changes,  m  will  rise  to  a  height, 
hf,  much  less  than  h  owing  to  the  decrease  of  kinetic  energy  that  takes 
place  when  the  moment  of  inertia  changes.  Moreover,  the  work  done 
against  friction  will  be  less  in  this  case  since  the  total  amount  of  ro- 


WOEK  AND  ENERGY 


161 


tation  will  be  less.  But  suppose  that  TO  at  the  lowest  point  of  its  de- 
scent is  replaced  by  a  smaller  mass  w',  such  that  the  moment  of  inertia 
having  changed,  TO'  is  finally  raised  to  the  height  h.  Then,  the  total 
rotation  being  the  same  in  the  two  cases  of  ascent  to  the  height  A,  the 
work  done  against  friction  will  be  the  same,  namely  W.  Hence,  if  the 
kinetic  energy  after  the  change  of  moment  of  inertia  be  E^  we  shall 
have  E2  =  m'yh  +  W.  Hence  El  —  E2  =  mgh  —  m'gh  =  (m  —  m'^)gh, 
m  —  m'  being  the  amount  by  which  m  was  supposed  decreased. 
The  magnitude  of  m  —  m'  can  be  ascertained  by  the  following 
arrangement. 

Let  m  be  replaced  by  two  scale  pans,  one  attached  below  the  other 
in  such  a  way  that  the  lower  one  becomes  detached  when  it  touches 
a  platform  placed  below  it.  (The  method  of 
attachment  is  indicated  in  Fig.  59.)  Let 
weights  be  placed  in  both  pans  and  let  the 
total  mass  (including  the  pans)  be  m.  The 
proper  distribution  of  weights  between  the 
two  pans  so  that,  the  lower  having  become 
detached  at  the  lowest  point  of  descent,  the 
upper  one  will  rise  again  to  the  height  h,  can 
be  found  after  a  few  trials.  A  little  thought 
will  show  how  after  but  one  trial  the  desired 
distribution  can  be  roughly  predicted.  A 
second  trial  with  this  predicted  value  will 

give  a  still  closer  approximation,  and  so  on.  

Four  or  five  such  trials  will  suffice  to  accom- 
plish the  object  sought. 

To  compare  this  experimental  value  of  El  —  E2  with  the  calculated 
value,  the  values  of  o^  and  co2  must  be  known.  These  may  be  taken 
from  the  results  of  Exercise  XVIII  provided  the  apparatus  and  condi- 
tions be  the  same.  To  test  the  latter  the  various  heights  H,  k,  h' 
should  be  tested  and  if  found  the  same  it  will  not  be  necessary  to  re- 
determine  Wj  and  <o2.  (It  may  be  noted  that  there  is  very  little  diffi- 
culty in  putting  up  the  apparatus  for  all  practical  purposes  exactly  as 
it  was  in  Exercise  XVIII.  The  only  difference  to  be  feared  is  in  the 
friction  of  the  bearings  and  this  friction  is  in  reality  very  small  com- 


FIG.  59. 


162  DYNAMICS 

pared  with  the  air -resistance  on  the  rotating  parts.)  The  times  oi 
ascent  and  descent  should  also  be  ascertained  and  noted. 

In  the  above  the  loss  of  kinetic  energy  has  been  found  experimen- 
tally, but  the  initial  and  final  values  of  the  kinetic  energy  can  also  be 
found.  First  for  El  we  have 

El  =  mgH  -  FH  (descent), 
El  =  mgh  +  Fh  (ascent), 

Hence  El  =  2  mg    Hh   . 

E2  can  be  found  in  a  similar  way  if  the  height  H'  from  which  m  must 
descend  in  order  that  it  may  reascend  to  the  height  h'  be  found  ex- 
perimentally 

E2  =  mgH'  -  F' H'  (descent), 

E2  =  mgh'  +  F'h'  (ascent), 


H'+h' 

Other  points  that  might  be  examined  experimentally  will  be  sug- 
gested below. 

DISCUSSION 

(a)    Sources  of  error. 

(&)    Suggestions  for  the  improvement  of  the  apparatus. 

(c)    What  weight  could  the  blocks  raise  while  sliding  out? 

(c?)  What  becomes  of  the  kinetic  energy  lost  on  change  of  moment 
of  inertia  ? 

(e)    Is  it  justifiable  to  assume  F  and  F'  equal  ? 

(/)  Can  F  and  F  be  found  directly  by  experiment? 

(#)  What  percentage  error  would  there  be  in  the  calculated  values 
of  El  and  E2  if  the  pendulum  were  not  exactly  a  second's  pendulum  ? 

(^)  Calculate  Wj  from  m,  the  time  of  descent  H,  and  the  radius  of 
the  axis.  Account  for  the  calculated  value  not  agreeing  exactly  with 
the  experimental  value. 

(i)  There  is  no  difficulty  in  arranging  the  apparatus  so  that  the 
sliding  blocks  are  released  when  only  half  of  the  thread  has  unwound. 
This  being  done,  what  will  the  velocity  be  when  the  whole  thread 
has  unwound? 

(j)    Analogy  between  this  exercise  and  the  one  on  impact. 


WORK  AND  ENERGY 


163 


In  Fig.  60  is  shown  an  apparatus  for  illustrating  qualitatively 
the  conservation  of  angular  momentum  when  moment  of  inertia  and 
angular  velocity  change  and  the  associated  changes  of  kinetic  energy 


FIG.  60. 


and  centrifugal  force.  The  vertical  brass  tube  that  supports  the 
horizontal  rod  is  carried  by  a  ball-bearing  and  the  cord  attached  to 
the  sliding  weight  is  connected  to  another  piece  of  tubing  which 
supports  a  spring  balance  by  a  ball-bearing  (both  bearings  may  be 


164  DYNAMICS 

made  from  a  bicycle  pedal).  If  the  sliding  blocks  are  massive 
compared  with  the  horizontal  rod,  the  changes  of  kinetic  energy  and 
centrifugal  force  may  be  stated  very  simply.  Let  m  be  the  mass  of 
each  block  and  r  its  distance  from  the  axis  of  rotation.  The  total 
angular  momentum  Jf=2mr2w;  the  total  kinetic  energy  E  =  rarV2, 
and  the  centrifugal  force  F  =  ma)2r.  Since  M  remains  constant  when 

r  changes,  by  eliminating  <o  we  find  that  Ex  —  ,  and  F<x—.     If  r  be 

r2  rB 

reduced  to  one-half  by  downward  pressure  on  the  ring  of  the  balance, 
F  will  be  increased  eightfold,  as  will  be  fairly  well  shown  by  the 
balance,  provided  the  horizontal  rod  be  well  lubricated.  A  horizontal 
scale  with  a  movable  pointer  may  be  attached  to  the  framework  just 
below  the  rotating  arm  in  order  to  indicate  the  position  of  the  blocks. 
The  sudden  changes  of  angular  velocity  and  of  centrifugal  force 
are  very  striking. 

115.  Notes  on  Some  Difficulties. — Kinetic  and  potential 
energy  are  equivalent.  One  can  be  changed  into  the  other. 
But  while  kinetic  energy  involves  motion,  potential  energy 
is  energy  of  bodies  which  may  be  relatively  at  rest.  It  is 
somewhat  difficult  to  understand  how  the  energy  of  some- 
thing at  rest  should  be  equivalent  to  energy  that  depends 
on  motion.  Some  progress  is,  however,  being  made  in  the 
direction  of  explaining  potential  energy  as  being  really 
kinetic  energy  of  particles  too  small  to  be  separately 
visible.  Thus  the  heat  energy  of  a  body  is  believed  to 
be  kinetic  energy  of  the  particles  of  the  body.  The 
potential  energy  of  a  spring  may  also  at  basis  be  kinetic 
energy  of  particles  separately  invisible,  but  attempts  at  a 
definite  explanation  have  hitherto  been  fruitless. 

That  the  potential  energy  of  an  elastic  solid  may  conceivably  be 
energy  of  "  concealed  motion "  may  be  illustrated  by  the  apparatus 
shown  in  Fig.  60.  Suppose  the  rotating  parts  covered  up  so  as  to 
be  invisible.  To  "  stretch  "  the  apparatus  by  pulling  on  the  cord, 


WORK  AND  ENERGY  165 

work  is  required,  and  the  apparatus  will  do  the  same  amount  of  work 
in  "  contracting,"  thus  imitating  to  a  certain  extent  the  action  of  a 
spring.  A  person  ignorant  of  the. mechanism  might  have  to  be  content 
to  describe  the  energy  of  the  apparatus  as  potential  energy,  whereas  it 
is  really  kinetic  energy  of  the  rotating  masses.  This  must  not  be  under- 
stood as  anything  more  than  a  crude  illustration  of  the  statement  that 
all  potential  energy  may  be  really  kinetic  energy  of  invisible  parts. 

The  kinetic  energy  of  a  body  is  calculated  from  its 
mass  and  velocity.  Now  the  velocity  of  a  body  means  its 
velocity  relatively  to  some  point  taken  as  origin  (§§  6, 18), 
usually  a  point  on  the  surface  of  the  earth.  Thus  by 
kinetic  energy  we  can  never  mean  anything  but  energy 
of  relative  motion  of  bodies.  The  choice  of  a  point  on 
the  surface  of  the  earth  seems  arbitrary.  Would  it  affect 
our  calculation  if  we  should  choose  the  centre  of  the  earth 
or  the  centre  of  the  sun  or  a  star  as  origin  ?  To  answer 
this  we  may  note  that  it  is  in  reality  only  with  changes  of 
kinetic  energy  that  we  are  concerned.  Consider  for  ex- 
ample the  impact  of  two  bodies  (Exercise  XXVI).  Their 
kinetic  energy  before  impact  may  be  considered  as  con- 
sisting of  two  portions  (§  107),  Uv  or  the  energy  of  their 
motion  relatively  to  their  centre  of  mass  Q  and  Ey  or  the 
energy  due  to  the  motion  of  0  relatively  to  the  point  taken 
as  origin.  The  impact  does  not  change  the  motion  of  C 
(§  85)  and  hence  does  not  alter  Ey  Thus  the  loss  of 
kinetic  energy  is  the  decrease  in  E^  which  is  independent 
of  the  choice  of  origin.  Similar  considerations  apply  to 
other  cases. 

REFERENCES 

Balfour  Stewart's  "  Conservation  of  Energy." 

Maxwell's  "  Matter  and  Motion." 

Daniell's  "  Principles  of  Physics,"  Chapter  IV. 


CHAPTER  IX 

PERIODIC  MOTIONS   OF  RIGID  BODIES 

116.  Angular  Simple  Harmonic  Motion. — Linear  S.  H.  M. 
is  a  vibration  in  a  line  according  to  the  law  a  =  constant 
xx.  A  rigid  body  free  only  to  rotate  may  have  a  closely 
analogous  motion.  For  instance,  the  balance-wheel  of  a 
watch  rotates  first  in  one  direction  and  then  in  the  oppo- 
site direction,  its  excursions  (when  the  motion  is  steady) 
being  confined  to  a  certain  angle. 

Angular  S.  H.  M.  may  be  defined  as  the  motion  of  a 
body  that  vibrates  through  an  angle  in  such  a  way  that 
the  angular  acceleration  «  is  always  opposite  to  and  pro- 
portional to  the  angular  displacement  0,  or  so  that 

*=-A-0,  (1) 

A  remaining  constant  throughout  the  motion. 

The  meaning  of  the  constant  A  can  be  found  by  con- 
sidering the  motion  of  a  point  P  in  the  body.  Let  0  be 
the  projection  of  the  axis  of  rotation.  Then  P  performs 

vibrations  in  an  arc  of  a  circle 
of  radius  r.  If  6  be  the  angular 
displacement  of  the  body  at  any 
moment,  the  arc  through  which 
P  is  displaced  from  its  mean  posi- 
ti0niB««r& 

If  a  be  the  angular  acceleration  of  P  at  any  moment, 
the  linear  acceleration  of  P  along  the  arc  is  a  =  ra  (§  30). 

166 


PERIODIC  MOTIONS   OF  RIGID  BODIES  167 

Hence  the  acceleration  and  displacement  of  P  are  con- 
nected by  the  relation 

a  x 

xT  =  —  A  '-> 

or  a  =  —  A  •  x. 

Now  let  us  suppose  that  a  point,  Q,  vibrates  in  a  straight 
line,  and  has  at  each  moment  the  same  displacement  from 
a  point  in  the  straight  line  as  P  has  along  the  arc  of 
vibration,  and  the  same  acceleration  along  the  straight 
line  as  P  has  along  the  arc.  Then  for  the  motion  of  Q 
we  have  a  —  —  Ax,  or  the  motion  of  Q  is  S.  H.  M.  The 
period  of  vibration  of  $,  of  P,  and  of  the  rigid  body  are 
evidently  equal,  say  T.  But  the  period  of  Q's  motion 
is  (§40) 


ra 

(2) 

Hence  the  angular  acceleration  and  angular  displacement 
of  the  rigid  body  are  connected  by  the  relation 


/27rV 
*=-(-f) 


.0.  (3) 


If  a  body  has  an  angular  acceleration  opposite  to  and 
proportional  to  its  angular  displacement,  its  motion  is 
angular  S.  H.  M.,  and  the  period  of  vibration  can  be  found 
by  means  of  (2)  or  (3). 

117.  Torsional  Pendulum.  —  If  a  body  attached  to  a  wire 
be  turned  through  an  angle  and  released,  it  will  perform 


168  DYNAMICS 

angular  vibrations.  The  motion  is  due  to  the  fact  that 
the  twisted  wire,  tending  to  untwist,  exerts  a  couple  on 
the  body  and  so  sets  the  body  in  rotation,  and  the  body, 
when  once  started,  tends  to  persist  in  its  motion  owing  to 
its  moment  of  inertia. 

Let  the  length  of  the  wire  be  ?,  and  let  the  couple 
applied  to  the  free  end  be  (7,  the  other  end  being  fixed  ; 
then  the  angle,  0,  through  which  the  wire  is  twisted,  is,  by 
Hooke's  Law  of  Elasticity  (see  §  57),  proportional  to  C\  it 
is  also  proportional  to  the  length  Z,  for  each  unit  of  length 

is  twisted  to  the  same  amount.     Hence  0oc  Cl,  or  0—  —  , 

l> 

where  r  is  a  constant  for  the  same  wire,  and  is  called  its 
constant  of  torsionj  T  may  also  be  defined  as  the  couple 
per  unit  length  per  unit  angle  required  to  twist  the  wire. 
The  couple  exerted  by  the  twisted  wire  is  equal  and  oppo- 

site to  0  or  it  equals  —  —  - 

(t 

Let  the  torsional  pendulum  be  displaced  through  an 
angle  0,  and  let  the  angular  acceleration  imparted  to  it 
by  the  wire  be  a,  then  (§71) 


and  «  =  —  -=•  0, 

-L  L 

I  being  the  moment  of  inertia  of  the  pendulum.     Now  for 
a^given  pendulum,  y  is  a  constant.     Hence  the  motion  is 

J.  L 

angular  S.  H.  M.,  and  the  period  of  vibration  is 


T=27T\- 


PERIODIC  MOTIONS  OF  RIGID  BODIES  169 

118.  Comparison  of  Moments  of  Inertia  by  the  Torsional 
Pendulum.  —  It  follows  from  the  last  section  that  if  two 
bodies  are  hung  in  succession  from  the  same  wire,  and  if 
their  respective  moments  of  inertia  are  I  and  1^  and  their 
periods  of  torsional  vibration  T  and  Tv  then 

i  =  ^L 

I      T2' 

If  the  period  is  T  when  a  body  of  moment  of  inertia  I 
is  attached,  and  T±  when  to  this  body  is  attached  another 
of  moment  of  inertia  i,  then 

I+i     T? 
I     =  T* 


This  suggests  an  experimental  method  of  finding  the 
moment  of  inertia  of  a  body,  no  matter  how  irregular  it 
may  be  in  form. 

Exercise  XXVIII.     The  Torsional  Pendulum.     Comparison  of 
Moments  of  Inertia 

The  upper  end  of  a  vertical  wire  is  held  in  a  clamp,  and  the  lower 
end  is  attached  to  an  axis  that  passes  through  the  centre  of  a  right- 
angled  block  of  wood  and  is  perpendicular  to  one  pair  of  faces  of  the 
block. 

To  fix  the  position  of  the  block  when  it  is  at  rest,  fasten  a  pin  in 
the  under  surface  near  an  end,  and  adjust  a  support  in  which*is  an 
upright  pin  until  the  two  pins  are  in  line  when  the  block  is  at  rest. 
Find  by  means  of  a  stop-watch,  or  by  counting  the  ticks  of  a  clock  or 
chronometer,  the  time  required  for  a  number  of  oscillations  of  the 
pendulum.  If  a  chronometer  circuit  (foot-note  p.  86)  is  used,  begin 
counting  seconds  after  a  silence  of  the  relay,  and  note  the  nearest 
second  and  fifth  of  a  second  at  which  the  block  passes  its  position  of 


170 


DYNAMICS 


rest;  count  the  oscillations  until  the  next  silence  of  the  relay,  and 
again  note  the  time,  to  a  fifth  of  a  second,  when  the  block  passes 
through  its  position  of  rest.  The  observations  of  the 
time  of  vibration  should  be  repeated  a  number  of  times 
and  the  average  taken.  Then  place  two  equal  lead 
cylinders  of  known  mass  on  the  block  so  that  the 
centres  of  the  cylinders  are  at  equal  distances  from 
the  axis  of  oscillation,  and  find  the  time  of  oscillation 
as  before. 

From  these  observations,  together  with  the  masses 
and  radii  of  the  lead  cylinders  and  their  distances  from 
the  axis  of  oscillation,  the  moment  of  inertia  of  the 
block  can  be  calculated  (§  118).  The  moment  of 
inertia  should  also  be  found  by  direct  calculation  from 
the  formula  proven  in  §  74. 

The  constant  of  torsion  of  the  wire  can  be  readily 
deduced  from  the  above  results,  and  then  the  moment 
of  inertia  of  any  other  body  can  be  found  by  attaching 
it  to  the  same  wire  and  finding  the  time  of  vibration. 
This  method  may  be  applied  to  find  the  moment  of  inertia  of  a  circular 
cylinder  about  an  axis  at  right  angles  to  its  geometrical  axis.  This 
will  afford  a  test  of  the  formula  proven  in  §  82.  Or  the  moment  of 
inertia  of  a  sphere  of  wood  may  be  found  and  the  formula  /  =  f  MR2 
tested. 

DISCUSSION 

(a)  Meaning  and  proof  of  formula  used. 
(&)  Effect  of  errors  in  adjustment : 

1.  If  the  line  of  the  wire  be  2  mrn.  from  the  centre  of  the  block. 

2.  If  the  block  be  inclined  at  2°  to  the  horizontal. 

(c)  How  much  error  would  result  from  supposing  that  the  lead 
cylinders  acted  as  if  concentrated  at  their  centres  ? 

(d)  In  the  case  of  an  ordinary  pendulum  the  arc  of  vibration  must 
be  small.     Need  this  be  so  in  the  case  of  a  torsional  pendulum  ? 

(e)  Might  a  bifilar  suspension  (two  parallel  vertical  cords)  be  used 
instead  of  a  wire,  without  any  change  in  the  calculation  ? 


FIG.  62. 


PERIODIC  MOTIONS   OF  EIGID  BODIES 


171 


119.  The  Compound  Pendulum.  —  A  rigid  body  vibrating 
under  the  influence  of  gravity  about  a  fixed  horizontal 
axis  is  called  a  compound  or  physical  pen- 
dulum. Let  S  be  the  projection  of  the  axis 
of  suspension  on  a  vertical  plane  through 
the  centre  of  gravity  O.  Let  SO  be  denoted 
by  A,  and  let  the  pendulum  be  displaced 
through  an  angle  6 ;  then  the  perpendicular 
from  S  on  the  vertical  line  through  0  is 
equal  to  h  sin  0.  Hence,  if  mg  is  the  ver- 
tical force  of  gravity  acting  at  the  centre  of 
gravity  of  the  body,  the  moment  of  gravity  about  S  is 
—  mgh  sin  0,  negative  when  6  is  positive,  and  vice  versa. 
If  /  is  the  moment  of  inertia  of  the  pendulum  about  the 
axis  S,  and  a  its  angular  acceleration, 

—  mgh  sin  0  =  Z«, 


or 


fmgh\  a 

—  (-T-M, 


if  0  be  a  small  angle.  Hence,  for  small  angles  of  vibra- 
tion the  motion  is  angular  S.  H.  M.,  and  the  period  of 
vibration  is  (§  116) 


If  the  radius  of  gyration  about  an  axis  through  the 
centre  of  gravity,  parallel  to  the  axis  of  suspension,  is  &0, 


and 


(a)  Equivalent  Simple  Pendulum.  —  If  the  above  for- 
mula be  compared  with  the  formula  for  the  time  of  vibra- 


172  DYNAMICS 

tion  of  a  simple  pendulum  (§  44),  it  will  be  seen  that  the 
compound  pendulum  vibrates  in  the  same  time  as  a  simple 
pendulum  whose  length  is 

,_*„*  +  *» 

A 

(5)  Centre  of  Oscillation.  —  It  follows  from  the  above 
that  the  compound  pendulum  vibrates  as  if  its  whole 
mass  were  concentrated  at  a  point  0  in  jSC  such  that 


h 

The  point  0  is  called  the  centre  of  oscillation,  correspond- 
ing to  S,  which  is  called  the  centre  of  suspension.  Since 
I  >  A,  S  and  0  are  on  opposite  sides  of  O. 

(c)  Centre   of  Suspension  and   Oscillation  Interchange- 
able. —  The  equation  for  I  may  also  be  written 

k*=h(l-h°)=SC-  OC. 

If  the  pendulum  be  suspended  so  as  to  vibrate  about  an 
axis  through  0,  parallel  to  the  original  axis  of  suspension 
through  $,  then  the  new  centre  of  oscillation,  $',  will  be 
found  by  a  similar  equation,  namely,  k^=  OC  •  S'C.  If 
these  two  equations  be  compared,  it  will  be  seen  that  81 
must  coincide  with  S.  Hence  a  pendulum  vibrates  in  the 
same  time  about  an  axis  through  any  centre  of  suspension, 
and  about  a  parallel  axis  through  the  corresponding  axis 
of  oscillation,  or  briefly,  any  centre  of  suspension  and  the 
corresponding  centre  of  oscillation  are  interchangeable. 

(d)  The  Reversible  Pendulum.  —  A  form  of  pendulum 
used  for  very  accurate  measurements  of  g  is  founded  on 
the  principle  just  stated.     It  consists  of  a  rigid  rod  pro- 
vided with  two  parallel  axes  of  suspension  in  the  form  of 


PERIODIC  MOTIONS   OF  RIGID  BODIES  173 

knife-edges.  These  axes  are  at  right  angles  to  the  rod, 
on  opposite  sides  of  the  centre  of  gravity,  and  in  a  plane 
passing  through  the  centre  of  gravity.  The  position  of 
the  centre  of  gravity  can  be  varied  by  one  or  two  weights 
movable  along  the  rod.  If  the  positions  of  the  weights  be 
adjusted  so  that  the  times  of  vibration  of  the  pendulum 
about  the  two  axes  are  equal,  then  the  length  of  the  equi- 
valent simple  pendulum  is  the  distance  between  the  knife- 
edges  (provided  they  be  not  equidistant  from  the  centre 
of  gravity  —  see  below).  Thus,  as  in  the  case  of  the 
simple  pendulum,  only  two  quantities,  I  and  T,  need  be 
determined.  (Poynting  and  Thomson,  page  12.) 

(e)  Parallel  Axes  about  which  the  Times  of  Vibration 
are  Equal.  —  The  radius  of  gyration  about  a  certain  axis 
through  the  centre  of  gravity  being  &0,  what  is  the  dis- 
tance from  the  centre  of  gravity  of  a  parallel  axis  about 
which  the  pendulum  vibrates  as  a  simple  pendulum  of 
length  I  ?  To  answer  this  we  must  find  the  value  of  h 
that  will  satisfy  the  equation  hz  —  hl  +  &02  =  0,  I  and  7cQ 
being  given.  The  solution  is 


Hence,  if  I  is  greater  than  2  &0,  there  are  two  values  of  h 
that  satisfy  the  conditions,  and  their  sum  is  I.  But  noth- 
ing has  been  specified  as  to  the  direction  in  which  h  is  to 
be  measured  from  the  centre  of  gravity.  Hence  all  the 
parallel  axes  about  which  the  pendulum  vibrates  in  the 
same  time  pass  through  two  circles,  and  the  sum  of 
the  radii  of  the  circles  equals  the  length  of  the  equiva- 
lent simple  pendulum.  But  the  length  of  the  equivalent 
simple  pendulum  also  equals  the  distance  between  a  centre 


174 


DYNAMICS 


of  suspension  and  the  corresponding  centre  of  oscillation. 
Hence,  as  the  centre  of  suspension  travels  around  one  of  the 
circles,  the  centre  of  oscillation  travels  around  the  other. 
(The  reader  should  interpret  the  solution  of  h2  —  Ih 
+  &02  =0  when  I  =  2  7c0,  and  when  I  <  2  &0.) 

(/)  Curve  of  h  and  L  —  At  various  points  along  a  line 
AB  (Fig.  64),  passing  through  the  centre  of  gravity  of 
the  pendulum,  suppose  parallel  axes  of  vibration  fixed  in 

the  body  perpendicular 
to  AB.  Let  the  length, 
Z,  of  the  equivalent  sim- 
ple pendulum  corre- 
sponding to  each  one  of 
these  axes  be  deter- 
mined experimentally. 
Then  assume  AB  as  an 
axis  of  abscissae,  and 
the  centre  of  gravity  as 
origin,  and  plot  a  curve 
with  the  values  of  h  as 
abscissae,  and  the  corresponding  values  of  I  as  ordinates. 
This  curve  will  show  at  a  glance  the  general  relation  be- 
tween h  and  L  The  form  of  this  curve  could  have  been 
predicted.  For,  corresponding  to  any  value  of  I  (above  a 
certain  limit  2&0),  two  values  of  h  can  be  found  on  each 
side  of  the  centre  of  gravity,  the  two  smaller  values  being 
equal,  and  likewise  the  two  larger  values.  Hence  for  any 
given  value  of  I  there  are  four  points,  P,  $,  R,  S,  on  the 
curve,  and  PR  =  QS  =  I. 

(#)  Graphical  Method  of  solving  k^=h(l—Ji).  —  On 
the  line  AB,  referred  to  in  (/),  erect  at  the  centre  of 


\P               J 

\ 

R                   SX 

I                  I 

/                   I 

^S'                   SCO'                  0" 

FIG.  64. 

PERIODIC  MOTIONS   OF  RIGID  BODIES 


175 


gravity  a  perpendicular,  CK=  JcQ.  With  a  pair  of  com- 
passes find  the  point  P  on  AB  such  that  KP  =  J  Z,  and, 
with  P  as  centre,  draw  through  K  a  circle  cutting  AB  in 
S  and  0. 


SO'. 

FIG.  65. 


Since  OK*  =  08-00  and  OS +00  =  I,  either  OS  or  CO 
represents  the  required  value  of  h.  Two  circles  can  be 
drawn  according  to  the  above  directions.  Hence  we  get 
fomr  points  on  AB,  about  which  the  pendulum  vibrates  as 
a  simple  pendulum  of  length  I.  If  this  diagram  be  rotated 
about  OK,  S  and  O  will  describe  the  two  circles  referred 
to  in  (e). 

By  reversing  the  construction  we 
can  find  &0  if  h  and  I  (i.e.  OS  and 
00)  are  known.  All  the  circles  like 
those  in  the  diagram,  drawn  with 
known  corresponding  positions  of  O 
and  S,  should  pass  through  K. 

(Ji)  Centre  of  Percussion.  —  If  the 
pendulum  be  at  rest,  at  what  point 

must  a  horizontal  force   be   applied     -^ 

to  it  so  that   it  will   start   without 
exerting  any  side-force  on  the  sup- 

,9      IT  Fra-66. 

port?     Here  we  are  concerned  only 

with  horizontal  forces  and  motions.     Hence  we  may  neg- 
lect gravity  and  suppose  the  body  free  in  space.      The 


176  DYNAMICS 

problem  then  is  to  find  where  F  must  be  applied  in  order 
that  when  motion  begins  S  may  remain  at  rest  as  if  fixed. 
The  moment  of  F  about  S  is  FV  ,  if  I'  is  the  distance  of  0', 
the  point  of  application  of  F,  from  S.  Hence  if  k  be  the 
radius  of  gyration  about  S,  the  body  will  start  rotating 

Flf 
about  S  with  an  angular  acceleration  a  =  —  -~  (§71).    But 

a  force  F  applied  to  a  free  body  of  mass  ra  starts  the  centre 

F 

of  mass  with  a  linear  acceleration  a  =  —  (§  85).     Since  S 

remains  at  rest,  a  =  ha  (§  30). 

Hence  * 

m 


Thus  the  point  0,  called  the  centre  of  percussion,  coincides 
with  the  centre  of  oscillation. 

Exercise  XXIX.    The  Compound  Pendulum 

A  simple  form  of  compound  pendulum  that  will  suffice  for  the 
present  purpose  is  a  brass  bar  pierced  by  several  holes  and  swinging 
on  a  knife-edge  that  passes  through  one  of  the  holes.  For  finding 
the  length  of  the  equivalent  simple  pendulum  a  simple  pendulum  of 
adjustable  length  is  hung  from  the  same  knife-edge. 

The  centre  of  gravity  of  the  bar  may  be  found  by  balancing  it 
across  the  knife-edge.  The  point  should  be  marked  by  a  lead  pencil. 
To  prevent  confusion  one  end  of  the  bar  may  be  lettered  A  and  the 
other  B,  and  the  holes  numbered  from  one  end  of  the  bar  to  the 
other.  The  distance,  A,  from  the  centre  of  gravity  to  that  point  on 
the  circumference  of  each  hole  which  will  be  in  contact  with  the 
knife-edge  should  be  carefully  measured  and  the  results  tabulated. 

Suspend  the  pendulum  on  the  knife-edge  and  find  the  length  I  of 
the  equivalent  simple  pendulum  for  each  position  of  the  axis  of  sus- 


PERIODIC  MOTIONS   OF  EIGID  BODIES 


177 


pension  and  tabulate  the  results.     Similar  observations  may  be  made 
for  axes  outside  of  the  bar  by  attaching  a  cord  to  the  end  of  the 
bar  and  swinging  the  bar  from  the  end  of  the 
cord. 

To  represent  these  results  graphically,  draw  a 
curve  with  values  of  h  (positive  toward  A  and  nega- 
tive toward  B)  as  abscissae  and  values  of  I  as  ordi- 
nates.  Test  the  curve  by  measurements  between 
the  two  branches  as  indicated  in  (/)  and  tabulate 
the,  results. 

Some  observations  of  the  length  of  the  equiva- 
lent simple  pendulum  for  axes  that  do  not  intersect 
the  axis  of  the  bar  may  be  made  by  tying  the  ends 
of  a  cord  to  holes  in  the  bar  and  hanging  the  cord 
over  the  knife-edge.  Values  of  I  should  be  ob- 
tained for  three  or  four  such  positions  of  the  axis 
of  suspension  and  the  statements  in  (e)  verified. 
Finally  £0  should  be  deduced  graphically  by  the 
method  suggested  in  (g). 

DISCUSSION 

(a)  Meaning  and  proof  of  formulae. 
(6)  Minimum  length  of  equivalent  simple  pen-  pIG  57 

dulum. 

(c)  Position  of  axes  about  which  the  time  of  vibration  is  a  minimum. 

(d)  Use  of  a  compound  pendulum  for  accurate  determinations  of  g. 

(e)  Where  should  a  base  ball  strike  the  bat  to  cause  no  jar  on 
the  hands? 

120.  Effect  of  an  Angular  Velocity  about  One  Axis  and  an 
Angular  Acceleration  about  an  Axis  at  Right  Angles  to  the  First. 
The  Gyroscope.  —  We  have  already  had  several  illustrations 
of  an  analogy  between  the  motion  of  translation  of  a  particle 
and  the  motion  of  rotation  of  a  rigid  body  (§§  30,  71, 116). 
This  analogy  between  rotation  and  translation  extends  still 


178  DYNAMICS 

further  and  we  shall  consider  one  interesting  example. 
The  motion  of  a  particle  that  has  a  constant  speed  and  a 
constant  acceleration  at  right  angles  to  the  speed  is  a  uni- 
form circular  motion  (§  33).  What  is  the  effect  of  a 
constant  angular  speed  about  one  axis  and  a  constant  an- 
gular acceleration  about  another  axis  at  right  angles  to 
the  first  ?  By  analogy  we  should  expect  the  result  to  be 
a  constant  revolution  of  the  axis  of  rotation. 

This  is  the  problem  of  the  Gyroscope  which  in  its  sim- 
plest form  consists  of  a  heavy  wheel  (of  mass  m)  rotating 
with  angular  speed  (o>)  about  a  horizontal  axis  (00)  and 

supported  at  a  point  (0)  in  the 
axis  of  rotation.  A  full  discussion 
of  this  instructive  and  interesting 
apparatus  would  be  far  beyond 
the  scope  of  this  book.  The  fol- 
lowing brief  and  incomplete  ac- 
count will  at  least  suggest  the  chief 

TTir*     f\& 

characteristics  of  the  motion. 

Let  us  first  suppose  that  the  wheel,  while  rotating  about 
its  axis  00  with  a  constant  angular  velocity  o>,  is  kept 
revolving  about  the  axis  0  V  with  an  angular  velocity  cor 
and  inquire  what  moment  of  force  is  necessary  to  keep  up 
the  revolution  about  0V.  That  some  moment  of  force  is 
necessary  is  evident,  for,  while  the  angular  momentum 
about  0V  is  constant  in  both  magnitude  and  direction,  the 
angular  momentum  Ico  about  00  is  constant  in  magnitude 
but  changes  steadily  in  direction  as  00  revolves.  (Com- 
pare the  change  of  direction  of  momentum  as  a  particle 
revolves  in  a  circle.)  After  a  short  time  £,  00  will  have 
turned  into  a  position  OR  where  the  angle  OOR  =  a>'t. 


PERIODIC  MOTIONS   OF  RIGID  BODIES  179 

If  the  angular  momentum  of  the  wheel  be  represented  by 
00  in  its  first  position  and  by  OR  in  its  position  after 
time  t,  00  and  OR  will  be  equal  in  length  and,  if  the 
parallelogram  OORZ\)Q  completed,  OZ  will  represent  the 
change  of  angular  momentum.  (The  angular  momentum 
about  0  V,  being  constant,  suffers  no  change.)  This  change 
will  require  a  moment  of  force,  say  L,  about  OZ,  and, 
since  the  change  is  produced  in  time  £,  the  angular  mo- 
mentum represented  by  OZ  must  equal  Lt.  Hence  OZ 
and  00  are  proportional  to  Lt  and  I(o  respectively.  Now 
the  small  angle  OOR  or  co't  may  be  taken  as  equal  to 
OR  +  00  or  OZ+  00. 

r*     Lt 
.•.   co't  =  — 

1(0 

or  L  =  Icoco1. 

Hence  for  steady  revolution  about  0  V  the  wheel  must  be 
acted  on  by  a  force  that  has  no  moment  about  OJ^or  00 
but  has  a  moment  Icoco'  about  an  axis  OZ  always  at  right 
angles  to  OP^and  00.  (Compare  the  formula  for  centrif- 
ugal force,  §  64,  written  in  the  form  F  =  rnvco'  where  cor 
or  v  -r-  r  is  the  rate  at  which  the  direction  of  mv  rotates.) 
Now  the  weight  of  the  wheel  acting  at  its  centre  of  mass 
at  a  distance  h  from  0  supplies  precisely  such  a  moment, 
mg~h,  about  OZ.  Hence  if  the  wheel  be  started  with  an 


angular  velocity       ~  about  0  V  it  will,  under  the  action 

1(0 

of  gravity,  continue  to  revolve  about  0V  with  that  angu- 
lar velocity.  Such  a  motion  is  called  precession.  If  the 
rotating  wheel  be  merely  released  without  any  angular 
velocity  about  0V  being  imparted  to  it,  gravity  will  at 


180  DYNAMICS 

first  cause  a  fall  of  the  centre  of  gravity,  but,  since  this 
will  be  accompanied  by  angular  momentum  about  OZ, 
precession  will  set  in  and  will  continue  at  a  rate  given  by 
the  above  formula  but  oscillations  similar  to  those  of  a 
badly  thrown  quoit  will  accompany  the  precession. 

The  necessity  for  an  initial  impulse  about  0  V  may  also 
be  stated  in  the  following  way.  Precession- about  0V im- 
plies an  angular  momentum  about  0V.  If  this  .be  not 
supplied,  the  start  will  be  opposed  by  inertia  in  the  form 
of  moment  of  inertia  of  the  wheel  about  0V.  The  effect 
of  inertia  will  be  equivalent  to  that  of  an  opposing  moment 
of  force  about  an  axis  0  V  drawn  vertically  downwards, 
and  the  effect  of  this  will  be  to  depress  the  axis  0  0.  The 
same  effect  follows  any  attempt  to  oppose  the  precession  of 
the  wheel  when  once  started,  whereas  an  attempt  to  ac- 
celerate the  precession  causes  an  elevation  of  OO. 

If  T  be  the  period  of  precession,  i.e.  the  time  of  one 
complete  revolution  about  0  F,  then 

Te>'  =  2  TT. 
i 

T~ 

Hence  T= 


mgh 

The  rotating  armature  of  a  dynamo  on  a  ship  that  is  roll- 
ing or  pitching  acts  like  a  gyroscope.  If  the  axis  of  the 
armature  be  at  right  angles  to  the  length  of  the  ship  and 
the  ship  be  rolling  with  an  angular  velocity  o>'  while  the 
armature  of  moment  of  inertia  /  is  revolving  with  angular 
velocity  &>,  then  the  bearings  must  supply  the  horizontal 
couple  1(0(0' ' .  If  the  distance  between  the  bearings  be  #, 

the  horizontal  pressure  on  each  bearing  will  be-— •• 


PERIODIC  MOTIONS  OF  RIGID. BODIES 


181 


Exercise  XXX.    The  Gyroscope 

Apparatus.  —  A  steel  rod  is  attached  by  a  double  nut  to  the  axle  of 
a  bicycle  wheel ;  this  rod  passes  loosely  through  a  hole  in  the  verti- 
cal steel  axis  used  in  previous  exercises,  being  carried  by  a  pin  that 


FIG.  69. 


passes  through  rod  and  axis.  The  pin  fits  tightly  into  the  rod  and  its 
ends,  which  are  ground  to  knife-edges,  rest  loosely  in  the  holes  in  the 
vertical  axis  so  that  the  rod  is  free  to  vibrate  in  a  vertical  plane.  A 
weight  to  counterpoise  the  wheel  can  be  clamped  on  the  rod  and  a 
smaller  movable  weight  can  be  clamped  at  any  desired  point  on 


182  DYNAMICS 

the  rod  so  as  to  produce  a  moment  of  force  about  the  line  of  the 
knife-edges  of  the  pin.  The  wheel  is  started  into  rotation  about  its 
own  axis  by  means  of  a  thread  that  is  wrapped  around  the  hub  of  the 
wheel  and  carries  a  weight,  the  end  of  the  rod  being  meanwhile  held. 
The  necessary  initial  impulse  may  be  given  with  the  hand,  but  a  more 
satisfactory  means  is  the  simple  starting  device  shown  in  the  accom- 
panying figure.  A  hinge  turning  about  a  vertical  axis  is  attached  by  a 
cord  to  a  spring  the  tension  in  which  can  be  varied.  The  hinge  being 
in  the  position  shown  in  the  figure,  a  loop  of  wire  attached  to  the 
hinge  encircles  the  end  of  the  rod.  If  the  hinge  be  released  the  spring 
will  rotate  it  and  start  the  gyroscope  with  an  impulse  that  depends  on 
the  tension  of  the  spring.  The  hinge  can  be  clamped  by  means  of  a 
hook  and  released  by  a  jerk  on  a  cord  attached  to  the  hook.  When 
drawn  aside  by  the  spring  the  hinge  will  not  be  in  the  way  of  the  rod 
as  it  returns  after  a  precession  of  the  gyroscope.  The  block  on  which 
the  hinge  is  mounted  can  be  turned  through  180°  in  order  to  start  the 
gyroscope  precessing  in  the  opposite  direction. 

While  the  counterpoise  and  the  small  weight  afford  a  means  of 
greatly  varying  the  action  of  the  gyroscope,  it  will  be  well  in  making 
a  first  quantitative  test  to  discard  both  and  use  the  instrument  in  its 
simplest  form.  What  immediately  follows  applies  to  this  case. 

Calculation  of  T.  —  To  calculate  the  period  of  precession,  the  pro- 
duct, /w,  and  the  moment  mgh,  must  be  known.  The  angular  momen- 
tum, 7o),  of  the  wheel  can  be  calculated  from  the  mass,  mf,  of  the 
descending  weight,  the  radius,  r,  of  the  hub,  and  the  time  of  descent ; 
for  if  a  be  the  angular  acceleration  of  the  wheel, 

m'gr  =  la, 
and  if  t  be  the  time  during  which  the  thread  is  attached  to  the  wheel, 

m'grt  =  lat  =  /«. 

This  is  on  the  assumption  that  the  frictional  resistance  to  the  rotation 
of  the  wheel  is  negligible.  This  resistance  is  small,  but  a  rough  esti- 
mate of  it  can  be  made  by  finding  what  small  weight  m"  attached  to 
the  thread  will  keep  the  wheel  in  steady  rotation.  Before  7o>  is 
calculated,  m"  must  be  subtracted  from  m'. 

The  time  of  descent  of  m'  should  be  ascertained  with  all  the  care 
possible.  This  should  be  done  several  times,  m'  being  always  started 


PERIODIC  MOTIONS  OF  RIGID  BODIES  183 

from  the  same  height.  The  length  of  the  thread  should  be  such  that 
m'  will  reach  the  floor  as  nearly  as  possible  at  the  moment  when 
the  thread  becomes  detached  from  the  wheel.  The  wheel  should  be 
released  exactly  on  a  tick  of  the  clock.  The  succeeding  ticks  should 
be  counted,  and  the  time  at  which  in'  strikes  the  floor  estimated  to 
one-fifth  of  a  second.  A  strong  effort  should  be  made  to  have  the 
separate  determinations  as  independent  as  possible. 

The  value  of  mgh  cannot  be  calculated  directly  since  h  is  not 
readily  measured.  A  simple  and  accurate  method  is  to  hang  a  weight 
from  the  axis  by  a  thread  with  a  loop  that  slips  on  the  axis  and  adjust 
the  position  of  the  thread  until  the  wheel  is  counterbalanced.  The 
moment  of  the  weight  is  readily  found  and  that  of  the  wheel  is 
equal  to  it  but  with  the  opposite  sign.  For  accuracy  the  distance  of 
the  thread  from  the  knife-edge  should  be  large. 

Observation  of  T.  —  The  period  of  precession  should  next  be  ob- 
served several  times  with  similar  care.  The  mean  should  agree  with 
the  calculated  value  within  a  fraction  of  a  second. 

The  above  method  should  also  be  applied  to  one  or  two  cases  with 
the  counterpoise  and  small  weight  on  the  axis.  The  effect  of  attempt- 
ing to  accelerate  or  retard  precession  or  tilt  the  axis,  and  also  the 
nature  of  the  oscillation  that  accompany  precession  as  well  as  other 
points  suggested  by  §  119  should  be  examined. 

DISCUSSION 

(a)  Why  is  the  axis  of  the  wheel  gradually  depressed  as  precession 
continues  ? 

(6)  What  effect  does  the  frictional  retarding  force  of  the  air  and 
of  the  bearings  on  the  wheel  produce? 

(c)  What  determines  the  direction  of  precession  ? 

(d)  Is  the  period  of  precession  influenced  in  any  way  by  the  tension 
of  the  spring  and  the  strength  of  the  initial  impulse  it  produces  ? 

(e)  Why  is  the  impulse  required  of  the  spring  much  less  when  the 
counterpoise  is  discarded  ? 

(/)  Does  gravity  do  any  work  during  the  motion? 
(g)  Does  the  impulse  due  to  the  spring  in  any  way  determine  the 
period  of  oscillation  of  the  gyroscope  ? 


184  DYNAMICS 

(Ji)  Explain  the  motion  of  a  top. 

(f)  Precessional  motion  of  the  earth.  (Young's  "General  Astron- 
omy," §§  205-212.) 

(/)  A  method  of  finding  /  experimentally  by  swinging  the  wheel 
as  a  compound  pendulum. 

(k)  A  method  of  finding  o>  (/  being  found  separately)  from  the 
mass  of  m  and  its  distance  of  descent. 

(/)  Why  does  the  force  of  gravity  not  change  the  angular  velocity 
of  the  body  about  OC? 

(m)  If  the  wheel  be  not  quite  symmetrical  about  OC,  what  will  be 
the  nature  of  the  motion  ? 

(n)  The  armature  of  a  dynamo  on  a  ship  weighs  500  kilos.  Its 
axis  is  at  right  angles  to  the  length  of  the  ship,  and  the  radius  of 
gyration  is  50  cm.  If  the  armature  is  running  at  a  speed  of  500 
revolutions  per  minute,  and  the  distance  between  the  centres  of  the 
bearings  is  50  cm.,  what  is  the  pressure  on  the  bearings  when  the 
ship  is  rolling  at  the  rate  of  £  of  a  radian  per  second  ? 

REFERENCES 

Mach's  "  Science  of  Mechanics,"  Chapter  II. 
Perry's  "  Spinning  Tops." 
Worthington's  "  Dynamics  of  Rotation." 


ELASTIC    SOLIDS    AND 
FLUIDS 

CHAPTER  X 

MECHANICS  OF  ELASTIC  SOLIDS 

1210  Solids  and  Fluids.  —  A  solid  is  a  body  that  has  a 
definite  shape  even  while  it  is  acted  on  by  forces  which 
tend  to  produce  a  change  of  shape.  A  fluid  is  a  body 
that  continues  to  change  its  shape  so  long  as  it  is  sub- 
jected to  forces  which  tend  to  produce  a  change  of  shape. 

The  definition  of  a  solid  does  not  imply  that  the  shape 
of  a  solid  is  invariable.  Some  change  of  shape  always 
occurs,  when  deforming  forces  are  applied  to  a  solid  ; 
but  the  solid  assumes  a  new  shape,  which  it  maintains 
so  long  as  the  forces  remain  constant.  (Some  slight 
qualification  of  this  last  statement  is  necessary,  but,  for 
convenience,  we  shall  postpone  it.) 

122.  Strain.  —  Any  change  of  shape  or  volume  or 
change  of  both  shape  and  volume  is  called  a  strain.  A 
change  of  shape  without  any  change  of  volume  is  called 
a  shear.  For  example,  when  a  rod  is  twisted  through  a 
small  angle  a  small  part  of  the  rod  that  was  originally 
cubical  assumes  a  new  shape,  but  suffers  no  change  of 
volume.  A  change  of  volume  without  any  change  of 
shape  has  usually  not  received  any  special  name ;  but,  to 

185 


186  MECHANICS 

avoid  the  repetition  of  the  phrase  "  change  of  volume  with- 
out change  of  shape,"  we  shall  call  it  a  squeeze.  It  is 
illustrated  by  the  compression  of  a  cube  into  a  smaller 
cube  or  a  sphere  into  a  smaller  sphere.  A  dilatation  is 
a  negative  squeeze. 

Shears  and  squeezes  are  called  simple  strains.  Strains 
which  involve  changes  of  both  shape  and  volume,  e.g.  the 
strain  of  a  stretched  wire  or  a  bent  beam,  can  be  resolved 
into  shears  and  squeezes. 

Any  strain  which  is  of  the  same  kind  and  magnitude 
at  all  points  throughout  the  strained  body,  is  called  homo- 
geneous strain.  All  parts  of  a  stretched  uniform  wire  are 
similarly  affected  and  the  strain  is  homogeneous.  When 
a  rod  is  twisted,  the  strain  is  greater  near  the  surface 
than  near  the  axis  and  the  strain  is  non-homogeneous. 
The  same  is  true  of  a  bent  beam.  In  such  a  case  the 
strain  in  a  very  small  part  of  the  body  may  be  regarded 
as  homogeneous. 

123.  Numerical  Measure  of  a  Squeeze.  — When  a  squeeze 
is  homogeneous,  it  is  measured  by  the  proportion  in  which 
the  whole  or  any  part  of  the  body  changes  in  volume,  or, 
if  Vj  be  the  original  volume  of  any  part  of  the  body  and  vz 
the  volume  to  which  that  part  is  reduced,  the  measure  of 
the  squeeze  is  (vj  —  v2)  -*-  vr 

If  the  squeeze  is  not  homogeneous,  to  find  its  measure  at 
any  point  we  must  suppose  vl  to  be  the  volume  of  a  small 
part  surrounding  that  point,  and  the  measure  of  the  squeeze 
at  that  point  is  the  value  approached  by  (^  —  v2)  -5-  v1  as 
the  part  considered  is  taken  smaller  and  smaller  without 
limit. 


ELASTIC  SOLIDS  187 

124.  Numerical  Measure  of  a  Shear. — First  consider  how 
a  shear  may  be  produced.     Between  two  horizontal  boards 
place  a  large  cube  of  firm  jelly  (a  calf's  foot 

jelly  containing  some  glycerine  will  do  admir- 
ably). Give  the  upper  board  a  horizontal 
displacement  in  a  direction  parallel  to  one 
vertical  face  ABOD  of  the  cube.  Any  plane 
parallel  to  ABGD  is  called  a  plane  of  shear.  Each  square 
in  a  plane  of  shear  becomes  a  rhombus.  All  planes  in  the 
body  parallel  to  the  boards  move  parallel  to  the  boards,  and 
the  measure  of  the  shear  is  the  relative  displacement  of 
any  two  of  these  parallel  planes  divided  by  the  distance 
between  them.  In  the  figure  the  shear  is  Bb  -r-  AB.  If 
<£>  is  the  amount  by  which  a  right  angle  in  a  plane  of 
shear  changes  tan  cf>  =  Bb  -+•  AB  =  the  shear,  and  if  the 
shear  is  small,  $  may  be  taken  as  its  measure  instead 
of  tan  (f). 

125.  Elasticity.  —  The  property  that,  enables  a  body  to 
recover  from  strain  is  called  elasticity.     When  the  strain 
recovered  from  is  a  shear,  the  elasticity  is  called  elasticity 
of  form;   when  it  is  a  squeeze,  the   elasticity  is   called 
elasticity  of  volume.     Forms   of   elasticity  corresponding 
to   the  various  forms  of   compound   strain   have   not,  in 
general,  received  special  names. 

A  body  that  recovers  completely  from  any  form  of 
strain  is  said  to  have  perfect  elasticity  of  that  form  ;  if  the 
recovery  is  incomplete,  the  elasticity  is  said  to  be  imper- 
fect. Probably  no  solid  has,  in  reality,  perfect  elasticity 
of  any  kind  ;  but  many  solids  are  so  nearly  perfectly  elas- 
tic, when  the  strain  does  not  exceed  a  certain  amount 


J.88  MECHANICS 

called  the  elastic  limit,  that  they  may  for  many  practical 
purposes  be  regarded  as  perfectly  elastic. 

A  solid,  such  as  putty  or  lead,  that  has  a  very  small 
elastic  limit  when  sheared  is  said  to  be  plastic ;  a  perma- 
nent change  of  form  of  such  a  body  can  be  produced  by 
comparatively  small  forces. 

126.  Stress. — When  an  elastic  body  is  in  a  state  of 
strain,  there  are  internal  forces,  actions  and  reactions,  be- 
tween contiguous  parts  of  the  body.  As  a  somewhat 
rough  illustration  consider  the  state  of  a  book  when  press- 
ures are  applied  normally  to  the  covers.  Any  leaf,  A, 
presses  against  a  contiguous  leaf,  B,  with  a  certain  force, 
and  B  presses  back  against  A  with  an  equal  and  opposite 
force.  If  the  forces  applied  to  the  cover  be  tangential  in- 
stead of  normal,  the  leaf  A  will  exert  a  tangential  force 
on  the  leaf  B,  and  B  will  exert  an  equal  and  opposite  tan- 
gential force  on  A.  The  condition  of  a  pillar  that  sup- 
ports a  weight  is  somewhat  similar  to  that  of  the  book 
in  the  first  case,  and  the  condition  of  any  small  part  of 
a  twisted  rod  is  similar  to  that  of  the  book  in  the  second 
case. 

The  action  and  reaction  between  the  contiguous  parts 
of  a  strained  body  constitute  a  stress.  The  measure  of  the 
stress  is  the  magnitude  of  the  force  (action  or  reaction) 
per  unit  of  area.  In  some  cases  the  stress  is  equal  to 
the  external  force,  per  unit  of  area,  applied  to  the  body. 
When  a  solid  is  immersed  in  a  liquid  to  which  pressure 
is  applied,  the  pressure  per  unit  area  within  the  solid 
equals  the  pressure,  per  unit  of  area,  exerted  on  the  solid 
by  the  liquid.  When  a  wire  is  stretched  by  a  weight 


ELASTIC  SOLIDS  189 

attached  to  it,  the  tension  per  unit  area  of  cross-section 
of  the  wire  equals  the  weight  sustained  divided  by  the 
cross-section. 

In  many  cases  the  stress  cannot  be  measured  directly 
by  the  external  force.  This  applies  to  a  beam  that  is  bent 
by  a  weight ;  if  the  weight  is  increased,  the  stress  at  any 
particular  point  is  increased  in  the  same  proportion  ;  but 
the  stress  is  different  at  different  points,  and  so  it  cannot 
be  measured  by  the  magnitude  of  the  weight.  There 
may  be  a  stress  within  a  body  to  which  no  external  force 
is  applied.  An  iron  casting,  when  cool,  has  internal  strains 
and  stresses  even  when  no  external  force  acts  on  it,  and 
a  glass  vessel  when  heated  irregularly  may  break,  owing 
to  the  magnitude  of  the  internal  strains  and  stresses. 

127.  Hooke's  Law.  —  Careful  -experiments  have  shown 
that,  so  long  as  the  strain  in  an  elastic  body  is  within  the 
elastic  limit,  the  ratio  of  the  measure  of  the  stress  to  that 
of  the  strain  is  constant.  This  law  was  first  stated  by 
Robert  Hooke  (in  1676)  in  the  words  ut  tensio  sic  vis,  or, 
stress  is  proportional  to  strain.  Hooke  illustrated  this  law 
by  the  stretching  of  a  spiral  spring,  the  twisting  of  a 
wire,  the  bending  of  a  plank,  etc.  Some  of  these  we 
shall  consider  later. 

Striking  evidence  of  the  correctness  of  Hooke's  Law  is  afforded 
by  the  vibrations  of  an  elastic  body  such  as  a  tuning-fork.  The  fre- 
quency of  a  S.  H.  M.  depends  on  the  ratio  of  the  restoring  force  to  the 
displacement  (§  57),  and  if  this  ratio  changed,  the  frequency  would 
change,  and  the  pitch  of  the  note  given  by  the  tuning-fork  would  also 
change.  But  the  pitch  of  a  tuning-fork  remains  constant  although 
the  amplitude  of  the  vibrations  decreases.  Now,  the  displacement  of 


190  MECHANICS 

a  point  on  the  fork  is  proportional  to  the  strain,  and  the  restoring 
force  is  proportional  to  the  stress,  and  the  fact  that  the  pitch  remains 
constant  as  the  vibrations  die  away  shows  that  the  fork  obeys 
Hooke's  Law.  A  tuning-fork  of  definite  pitch  may  be  made  of  any 
ordinary  metal,  even  lead,  and  this  shows  that,  within  the  elastic 
limit,  all  ordinary  metals  obey  Hooke's  Law  with  great,  if  not  perfect, 
accuracy. 

128.  Moduli  of  Elasticity.  —  The  modulus,  or  measure, 
of  the  elasticity  of  a  body  corresponding  to  any  form  of 
strain,  is  the  ratio  of  the  measure  of  the  stress  to  that  of  the 
strain.  There  is,  therefore,  a  modulus  for  each  possible 
form  of  strain,  but  only  a  small-  number  need  be  specially 
considered.  Two  that  may  be  considered  as  the  principal 
moduli  are  the  shear  modulus  (also  called  the  simple 
rigidity)  and  the  bulk  modulus. 

The  bulk  modulus  is  the  ratio  of  the  squeezing  stress 
to  the  squeeze.  The  measure  of  the  squeezing  stress  is 
the  pressure  per  unit  of  area,  p,  within  the  body.  If  the 
squeeze  is  due  to  liquid  pressure  applied  to  the  body, 
the  squeezing  stress  also  equals  the  pressure  per  unit  of 
area,  jp,  in  the  liquid.  The  measure  of  the  squeeze  (§  123) 
is  (flj  —  v2)  -5-  vv  Hence,  denoting  the  bulk  modulus  by  k, 


The  reciprocal  of  k  is  called  the  coefficient  of  compressi- 
bility of  the  substance.  It  evidently  equals  the  proportion 
in  which  the  volume  is  decreased  when  unit  pressure  is 
applied  to  the  body. 

The  shear  modulus  is  the  ratio  of  the  shearing  stress 


ELASTIC  SOLIDS  191 

to  the  shear.  The  shearing  stress  is  measured  by  the 
tangential  force,  T,  per  unit  of  area  within  the  body. 
Denoting  the  measure  of  the  shear  by  <£  (§  124)  and  the 
shear  modulus  by  n, 

-I- 

129.  Torsion  of  a  Wire  or  Rod.  —  The  strain  at  any  point 
of  a  wire  or  rod  subjected  to  a  slight  twist  is  a  shear. 
Consider  a  very  small  cube  one  edge  of  which 
is  parallel  to  the  length  of   the  wire,  while  a 
second  edge  lies  along  a  radius  of  the  section  of 
the  wire,  and  a  third  is  part    of   the   circum- 
ference of   a  circle  coaxial  with  the  wire.     A 
consideration   of   Fig.   71  will   show   that   the 
strain  of  the  cube  is  similar  to  that  of  the  cube 
of  jelly  referred  to  in  §  124.     The  stress  is  also 
a  shearing  stress.     The  shear  and  the  shearing 
stress  may  be  found  by  considering  the  dimen- 
sions of  the  wire  and  the  twist  it  undergoes  when  a  known 
couple  is  applied  to  one  end,  the  other  end  being  clamped. 

The  relations  between  the  twist  of  a  wire,  the  dimen- 
sions of  the  wire,  and  the  couple  that  produces  the  twist 
can  be  found  by  experiment  or  calculated  by  theory. 
The  following  exercise  will,  in  a  rough  way,  illustrate 
the  experimental  method. 

Exercise  XXXI.    The  Torsion  of  a  Wire 

A  vertical  wire  is  clamped  at  both  ends  and  carries  a  horizontal 
disk  which  is  clamped  to  the  middle  of  the  wire.  The  recording  disk 
of  Exercise  XI Y  may  be  used  for  the  purpose,  the  aperture  being 
reduced  by  the  insertion  of  a  "  connector  "  (such  as  is  used  in  elec- 


192 


MECHANICS 


trical  circuits),  the  set  screw  of  which  will  serve  to  clamp  the  wire. 
A  protractor,  or  graduated  paper  circle,  is  fastened  to  the  upper  side  of 
the  disk,  and  a  bent  wire  attached  to  a  cross-bar  serves  as  an  index  in 
measuring  the  rotation  of  the  centre  of  the  wire.  Tangential  forces 
are  applied  to  the  disk  by  means  of  threads  which  pass  over  pulleys, 


FIG.  72. 

attached  to  the  framework,  and  carry  scale  pans  and  weights.  (In- 
stead of  the  scale  pans,  weights,  and  pulleys,  calibrated  springs  may 
be  used,  though  not  as  readily.) 

We  may  first  inquire  whether  the  angle,  0,  through  which  the 
middle  of  the  wire  is  twisted  is  proportional  to  some  power,  p,  of  the 
couple,  L,  applied  to  the  wire.  If  so,  0  =  e  •  Lp,  c  being  some  constant 


ELASTIC  SOLIDS  193 

that  does  not  change  as  L  and  0  vary.  Hence  log  6  =  log  c  +  p  log  L. 
If  couples  Lv  Ly  Ly  —  produce  twists  Ov  02,  03,  ••.,  and  if  the  values 
of  logZj,  log£2,  ...  plotted  as  ordinates  against  the  value  of  log  Ov 
log#2,  ...  as  abscissae  give  a  straight  line,  it  will  show  that  0  is  pro- 
portional to  some  power  of  L.  From  the  values  of  Lv  L2,  and  0V  02 
a  value  of  p  may  be  obtained  by  substituting  in  the  equation  last 
stated  and  eliminating  log  c. 

log^-logfl, 
log  Ll  -  log  Lz 

Other  values  of  p  may  be  obtained  from  pairs  of  corresponding  values 
of  L  and  9.  These  values  of  p  should  agree  as  well  as  could  be 
expected,  when  the  unavoidable  errors  of  observation  are  considered. 
In  a  similar  way  we  may  find  the  relation  between  the  twist  of 
the  wire  and  its  length,  L  being  kept  constant.  The  length  may  be 
changed  by  altering  the  position  of  the  bars  to  which  it  is  clamped. 

Finally,  by  using  three  or  more  wires  of  the  same  material  and 
length,  twisted  by  the  same  couple,  we  may  find  the  relation  between 
twist  and  radius.  In  this  case,  unless  carefully  selected  wires  are 
chosen,  the  results  will  probably  not  be  so  satisfactory  as  in  the  pre- 
ceding cases.  In  fact,  the  results  may  be  considerably  affected  by  a 
serious  error  of  method  (in  addition  to  the  errors  of  observation)',  namely, 
the  use  of  wires  that  do  not  consist  of  the  same  material  in  the  same 
state.  It  is  well  known  that  the  process  of  wire  drawing  has  consider- 
able effect  on  the  physical  state  of  the  material.  Divergences  between 
the  results  amounting  to  several  per  cent  may  be  found ;  but,  as  their 
causes  are  understood,  their  existence  need  cause  no  dissatisfaction 
with  the  results. 

Allowing  for  errors  both  of  observation  and  of  method,  the  results 
will  show  that 


DISCUSSION 

(a)  Do  the  observations  confirm  Hooke's  Law? 
(&)  Describe  the  nature  of  the  strain  in  the  wire, 
o 


194  MECHANICS 

(c)  Does  the  magnitude  of  the  strain  at  a  point  depend  on  the  dis- 
tance of  the  point  from  the  axis? 

(d)  Is  the  strain  everywhere  the  same  at  equal  distances  from  the 
axis? 

(e)  Where  will  fracture  begin  if  a  uniform  glass  rod  be  twisted  too 
much? 

(/)  From  the  twist  of  the  free  end  calculate  the  twist  of  any  other 
cross-section. 

(#)  Did  the  tension  on  the  wire  affect  the  experimental  results  ? 

(h)  What  is  the  nature  of  the  strain  in  a  stretched  spiral 
spring? 

(i)  What  is  the  magnitude  of  the  shear  of  a  cubical  part  of  the 
wire  (§  129),  if  the  side  of  the  cube  is  0.01  mm.,  the  length  of  the 
wire  1  m.,  and  its  radius  1  mm.,  supposing  the  centre  of  the  cube  half- 
way from  the  axis  to  the  surface  of  the  wire  and  the  twist  of  the 
whole  wire  30°? 

130.  Theory  of  Torsion  of  a  Uniform  Wire.  —  For  definite- 
ness  we  shall  suppose  that  the  ends  of  the  wire  are  sections  perpen- 
dicular to  the  axis  and  are  cemented  firmly  to  disks,  one  disk  being 
held  fixed  while  the  other  is  turned  about  the  axis  of  the  wire  through 
an  angle  6.  Let  the  length  of  the  wire  be  I  and  its  radius  R.  Con- 
sider two  normal  sections  unit  distance  apart.  One  is  turned  through 
an  angle  0  -r-  I  relatively  to  the  other.  Hence  at  a  distance  r  from  the 
axis  the  measure  of  the  shear  is  rO  -f-  Z,  and  that  of  the  shearing  stress 
nrO  -s-  I. 

Suppose  the  area  of  the  end  which  is  attached  to  the  rotated  disk 
to'  be  divided  up  into  a  large  number  of  small  areas  sv  s2,  •••,  their 
respective  distances  from  the  axis  being  rv  r2,  •••.  To  these  the  disk 
applies  tangential  forces  nr^^O  -r-  I,  nr^s^O  -=-  I,  •••.  The  tangential 
force  applied  to  each  small  area  is  perpendicular  to  that  radius  of  the 
end  section  that  passes  through  the  centre  of  the  area.  Hence  the 
sum  of  the  moments  of  these  forces  about  the  axis,  that  is,  the  couple 
applied  to  the  free  end,  is 


ELASTIC  SOLIDS  195 

Now  3*r2  is  the  moment  of  inertia  of  a  disk  of  the  same  form  as 
the  end  section  and  of  unit  mass  per  unit  area  (it  is  often  called  the 
moment  of  inertia  of  the  section).  Denoting  it  by  /, 

r_nOl 
'•—' 

For  a  circular  wire  I  =  J  irR2  -  R*  =  \  7r724  (§  75),  and  therefore 


To  get  the  constant  of  torsion  of  the  wire  or  the  couple  per  unit 
length  per  unit  angle  required  to  twist  the  wire  (§  117),  we  put  0  =  1 
and  I  ±:  1.  Hence 

T  =  I  TTTlR*. 

(From  the  results  of  the  last  exercise  calculate  w.) 

131.  Kinetic  Method  of  finding  n.  —  The  last  formula  of 
the  preceding  suggests  a  method  of  measuring  the  shear 
modulus  of  a  wire.     To  the  wire  a  body  of  known  moment 
of  inertia  is  attached  and  the  time  of  a  torsional  vibration 
is  observed  (§  117  and  Exercise  XXVIII).     This  gives 
the  value  of  r,  and  that  of  n   is   found   from  r  and  R. 
(Calculate  from  the  results  of  Exercise  XXVIII  the  con- 
stant of  torsion  and  the  shear  modulus  of  the  wire.) 

132.  Stretch  Modulus.  —  When   a   wire   of   length   I  is 

stretched  to  a  length  I  -f-  #,  the  measure  of  the  strain  is  -, 

l 

or  the  proportion  in  which  the  length  is  increased.    If  the 
whole  force  applied  to  stretch  the  wire  is  F,  and  the  area 

of  cross-section  of  the  wire  is  s,  the  measure  of  the  stress, 

W 

or  the  tension  per  unit  of  cross-section,  is  —  .      Hence  the 

W      T         W7 

stretch  modulus  is  —  -J-  -  or  —  .     The  stretch  modulus  is 
s       I        sx 


196  MECHANICS 

also  called  Young's  modulus  from  the  name  of  the  physi- 
cist, Thomas  Young,  who  first  defined  it  (1807). 

The  definition  of  the  stretch  modulus  suggests  a  direct 
method  of  measuring  it.  The  length  and  radius  of  the  wire 
are  carefully  measured,  and  then  the  stretch  produced  by 
hanging  a  known  weight  to  the  wire  is  carefully  observed. 

When  a  rod  is  shortened  by  longitudinal  pressure,  the 
strain  is  a  negative  stretch.  The  stress  is  in  this  case  a 
thrust  which  may  be  considered  as  a  negative  stretching 
stress.  The  value  of  Young's  modulus  obtained  from  the 
strain  and  stress  in  the  compressed  rod  is  not  appreci- 
ably different  from  the  value  found  when  the  strain  is  a 
positive  stretch. 

133.  Poisson's  Ratio.  —  The  stretching  of  a  wire  involves 
a  change  of  shape,  or  a  shear,  since  the  length  increases 
and  the  diameter  decreases.  Whether  a  change  of  volume 
also  occurs  can  be  found  by  careful  measurements  of  the 
changes  of  length  and  diameter. 

Let  the  initial  radius  be  r0  and  the  final  r,  and  let  ?0  and  I 
be  the  corresponding  lengths.  If  no  change  of  volume  took 
place,  TT  rflQ  would  equal  7rr2£,  or  r  and  I  would  be  con- 
nected by  the  relation 


Experiment  shows  that  for  no  substance  is  this  relation 
true,  but  that  in  all  cases 

'LV  (1) 


q  being^  a  constant  for  each  substance.     The  smaller  q  is, 
the  smaller  is  the  ratio  in  which  the  radius  decreases  for  a 


ELASTIC  SOLIDS  197 

given  stretch  ;  and  since  it  is  found  that  in  all  cases  q  is 
less  than  ^  (see  Table  in  Appendix),  a  stretch  is  always 
accompanied  by  an  increase  of  volume. 

The  constant  q  is  called  Poisson's  ratio.  To  explain 
why  it  is  called  a  ratio  let  us  suppose  that  the  stretch  is 
very  small,  so  that  I  =  1Q  +  x,  and  r  =  r0  —  z,  x  and  z  being 
very  small.  Substituting  in  (1)  and  expanding, 

jo 

or  l-^  =  ('l+^y  =  l_^|, 

rQ    \     y  i0 

squares  and  higher  powers  of  x  -5-  1Q  being  neglected. 

Hence  ^T^T  (2) 

Thus  q  may  be  defined  as  the  ratio  of  the  fractional 
decrease  of  radius  to  the  fractional  increase  of  length, 
both  being  supposed  indefinitely  small. 

Exercise  XXXII.    Young's  Modulus  and  Poisson's  Ratio 

A  rubber  cord  about  3  mm.  in  diameter  is  clamped  in  the  slitted 
end  of  a  long  vertical  screw  which  turns  in  a  nut  attached  to  a  hori- 
zontal bar.  Any  desired  number  of  turns  can  be  given  to  the  screw 
by  turning  a  lever  on  the  top  of  the  screw. 

Through  the  cord  two  fine  sewing  needles  are  thrust,  and  the  dis- 
tance between  them  is  read  on  a  vertical  mirror  scale. 

The  mean  diameter  of  the  cord  is  found  by  observing  the  move- 
ments of  two  silk  threads  that  wind  around  the  cord  as  the  screw  is 
turned.  Each  thread  is  attached  to  the  cord  by  a  small  loop  that 
passes  over  one  end  of  the  upper  needle.  The  direction  of  the  thread 
is  at  first  horizontal,  but  at  a  distance  of  1  cm.  from  the  cof  d  it  passes 
through  a  small  screw-eye  and  hangs  vertically  in  front  of  the  mirror 


198 


MECHANICS 


Zl 


scale.  A  small  bullet  suspended  from  the  thread  keeps  the  thread 
under  definite  tension  and  also  serves  as  an  index.  The  movement 
of  the  bullet  can  be  read  to  about  .2  mm.  The 
use  of  two  threads  eliminates  errors  which  would 
enter  if  the  cord  were  drawn  sidewise  by  the 
tension  of  a  single  thread.  Allowance  must  be 
made  for  the  slope  of  the  thread  on  the  rubber 
cord.  The  angle  of  slope  can  be  calculated  from 
the  pitch  of  the  screw.  The  screw  should  be 
given  three  or  four  turns  before  readings  are 
begun. 

About  200  g.  are  placed  in  a  scale  pan 
attached  to  the  cord.  The  positions  of  both 
ends  of  each  needle  are  observed  on  the  scale 
and  also  the  positions  of  the  lowest  part  of  each 
bullet.  The  screw  is  then  given  ten  or  more 
complete  turns,  and  all  of  the  above  readings 
repeated.  The  screw  is  next  turned  back  to 
its  initial  position  and  the  readings  again 
repeated.  The  load  is  then  increased  by  100  g. 
and  the  above  operations  are  repeated  to  ob- 
tain the  new  length  and  diameter. 

From  these  readings  Young's  modulus  can 
be  calculated.  For  the  value  of  the  cross- 
section,  the  mean  of  its  values  before  and 
after  the  addition  of  the  100  g.  should  be 
taken. 

To  obtain  Poisson's  ratio,  take  the  logarithms 
of  both  sides  of  (1)  of  §  133. 

log  rQ  -  log  r  =  q  (log  I  -  log  Z0). 

Both  constants  should  be  obtained   several 
times  with  different  loads  on  the  pan,  increasing 
FlG  73  by  steps   of   100  g.  until  the  rubber  shows  a 

decided  permanent  set.     In  calculating  Young's 
modulus,  use  for  the  cross-section  the  mean  of  its  values  before  and 


ELASTIC  SOLIDS 


199 


after  the  last  addition  of  100  g.     In  calculating  Poisson's  ratio,  take 
for  rQ  and  /0  their  values  before  the  first  100  g.  were  added. 

DISCUSSION 

(a)  Do  the  observations  confirm  Hooke's  Law  ? 

(6)  Meaning  of  Young's  modulus  and  Poisson's  ratio. 

(c)  Why  would  an  incorrect  value  for  q  be  obtained  if  it  were  cal- 
culated from  the  experimental  results  with  the  aid  of  equation  (2)  of 
§  133?  Would  this  apply  to  measurements  of  a  metallic  wire? 

((/)  What  do  your  results  show  as  regards  changes  of  volume  of 
rubber  when  stretched? 

(e)  Do  the  results  indicate  anything  as  regards  the  elastic  limit  of 
rubber  ? 

134.  Flexure  of  a  Uniform  Bar.  —  When  a  uniform  bar 
is  bent,  longitudinal  lines  of  particles  on  the  convex  side 
are  lengthened,  while  those  on  the 
concave  side  are  shortened.  Lines  \ 
on  a  certain  intermediate  surface, 
called  the  neutral  surface,  are  not 
changed  in  length.  A  plane  con- 
taining any  one  of  these  curved 
lines  is  called  a  plane  of  bending.  If  the  bending  is  slight, 
the  extensions  and  compressions  are  similar  to  those  of  a 
rod  which  is  stretched  or  shortened  by  a  longitudinal 
force,  as  in  the  measurement  of  Young's  modulus. 

The  relations  between  the  amount  of  bending,  the  di- 
mensions of  the  bar,  and  the  force  applied  to  the  bar  may 
be  found  by  experiment  or  calculated  by  theory.  The 
following  exercise  will  illustrate  the  experimental  method. 

Exercise  XXXIII.    Flexure  of  a  Bar 

Uniform  brass  bars  of  rectangular  cross-section  are  supported  in 
succession  on  two  knife-edges  and  various  weights  are  hung  from  the 


FIG.  74. 


200 


MECHANICS 


centre  of  each  bar.  The  depression  is  measured  by  a  lever,  one  end 
of  which  rests  on  the  bar  by  means  of  two  needle  points,  while  a  point 
near  that  end  is  supported  on  a  cross-bar  of  the  framework  by  means 
of  a  single  needle  point.  A  needle  in  the  farther  end  of  the  lever 
moves  down  along  a  vertical  millimetre  mirror  scale  as  the  centre  of 
the  brass  bar  is  depressed.  Thus  the  movement  of  the  centre  of  the 
bar  is  read  on  a  scale  enlarged  in  proportion  to  the  ratio  of  the  arms 
of  the  lever. 

To  find  how  the  depression,  z,  of  the  centre  of  the  bar  thus  loaded 
depends  on  (1)  the  force,  F,  applied  to  the  bar,  (2)  the  length,  I,  of 
the  bar,  (3)  the  width,  b,  of  the  bar,  and  (4)  the  depth,  d,  of  the  bar, 
proceed  as  in  Exercise  XXXII.  The  method  is  so  entirely  similar 


FIG.  75. 


that  it  need  not  be  restated.  The  actual  value  of  x  is  not  needed  for 
the  present  purpose ;  the  scale  readings  are  proportional  to  the  values 
of  x  and  are  sufficient.  But  for  another  purpose  the  actual  values  of 
x  are  required.  Hence  the  lengths  of  the  arms  of  the  lever  should  be 
measured.  The  full  length  of  the  lever  from  the  single  needle  point 
to  the  end  of  the  index  needle  may  be  measured  by  an  ordinary  scale. 
The  length  of  the  short  arm  may  be  found  by  a  micrometer  caliper 
or  by  placing  the  lever  on  a  finely  divided  scale. 

(For  this  exercise  bars  should  be  supplied,  three  or  more  of  which 
have  the  same  thickness  and  length  but  different  widths,  while  three 
or  more  have  the  same  width  but  different  thicknesses.  Thus  at  least 
five  bars  will  be  needed.  Bars  of  the  same  thickness  may  be  sawn 
from  the  same  sheet  of  brass.) 


ELASTIC  SOLIDS  201 

DISCUSSION 

(a)  Do  the  results  agree  with  Hooke's  Law  ? 

(6)  Regarding  the  bar  as  made  up  of  parallel  wires,  how  are  the 
various  wires  changed  in  length  ? 

(c)  Would  change  in  length  of  the  wires  account  for  the  change 
of  shape  of  the  bar? 

(rf)  Is  there  any  change  in  the  cross-section  of  the  bar  ? 

(e)  What  relative  motion  of  two  adjacent  cross-sections  takes  place? 

(/)  What  kind  of  vibrations  would  the  loaded  bar  perform? 

(#)  It  can  be  shown  by  mathematical  methods  that  the  depression 

of  the  bar  is  • •     From  this  and  the  observations  made  calcu- 

ix        T,^  4:Mbd8 

late  M. 

(A)  Deduce  from  the  formula  in  (#)  a  formula  for  the  depression 
produced  by  a  weight  attached  to  one  end  of  a  bar  that  is  clamped 
horizontally  at  the  other  end. 

135.  Relation  between  Elastic  Moduli.  —  Since  the  stretch 
of  a  wire  and  the  flexure  of  a  bar  involve  both  shears  and 
squeezes,  it  is  evident  that  there  must  be  a  relation  be- 
tween the  stretch  modulus,  M,  the  shear  modulus,  n,  and 
the  bulk  modulus,  k.     This  relation  (the  proof  of  which 
we  must  omit)  is 

M=    9Jcn   - 

The  value  of  n  for  some  substances,  such  as  india- 
rubber,  is  very  small  compared  with  that  of  k,  and  M  is, 
therefore,  nearly  equal  to  3  n. 

If  M  and  n  be  carefully  measured,  k  can  be  deduced. 
The  direct  measurement  of  k  is  difficult. 

136.  Potential  Energy  of  Strain.  — Work  is  done  in  strain- 
ing an  elastic  body,  and  the  strained  body  has  an  amount 
of  potential  energy  equal  to  the  work  done.     It  can  be 


202  MECHANICS 

shown  that  the  amount  of  this  potential  energy  per  unit 
volume  of  the  strained  body  is  one-half  the  product  of  the 
stress  by  the  strain.  As  an  example,  consider  a  wire  of 
cross-section  s  and  length  I  initially  under  no  tension,  and 
suppose  that  it  is  stretched  by  a  gradually  increasing  force 
to  a  length  I  +  a?,  the  force  at  this  length  being  F.  Since 
the  force  is,  at  each  stage  of  the  stretching, 
proportional  to  the  extension  (Hookers  Law), 
the  diagram  of  work  (§  100)  is  a  triangle,  and 
the  total  work  done  or  potential  energy  pro- 
duced is,  therefore,  |  Fx.  If  the  amount  of 
the  stretch  is  small  (as  in  the  case  of  a  metallic  wire 
stretched  within  the  elastic  limit),  the  volume  of  the 
stretched  wire  is  si.  Hence  the  potential  energy  per 

-i     TT  TI 

unit  volume  is •  -  •     Now,  —  is  the  force  per  unit 

2  s      I  s 

area,  that  is,  the  stress,  and  -  is  the  stretch  per  unit  length, 
that  is,  the  strain. 

137.  Imperfections  of  Elasticity.  —  So  long  as  the  strain 
of  a  body  is  within  the  elastic  limit,  a  curve,  plotted  with 
stresses  as  ordinates  and  strains  as  abscissse,  is  (at  least 
very  nearly)  a  straight  line,  as  we  have  seen  in  the  last 
three  exercises.  Further  stress  will  cause  the  strain  to 
increase  more  rapidly  than  the  stress,  and  the  curve  will 
become  concave  downwards.  Finally,  a  point,  called  the 
yield  point,  will  be  reached  at  which  the  strain  will  in- 
crease very  rapidly  and  the  material  will  cease  to  act  as  a 
solid  and  begin  to  flow.  If  the  stress  be  then  relaxed,  a 
large  permanent  set  will  remain.  The  stamping  of  a  coin 
is  a  striking  illustration, 


ELASTIC  SOLIDS  203 

In  many  cases  the  elastic  limit  is  somewhat  indefinite, 
and  the  curve  of  stress  against  strain  is  everywhere  con- 
cave downwards.  When  the  stress  is  grad- 
ually decreased,  the  curve  is  not  retraced,  but 
another  curve,  concave  upwards,  is  obtained. 
This  is  called  elastic  hysteresis. 

In  other  cases  (glass,  for  example),  when  a 
stress   is   produced   and   kept  constant,  the 
strain  does  not  reach  its  full  value  at  once,  but  continues 
to  increase  for  some  time.     When  the  stress  is  relaxed, 
the  strain  nearly  disappears,  but  a  slight  residual  strain 
remains,   which   only  slowly  disappears.     This   is  called 
elastic  lag. 

If  an  elastic  body  be  in  some  way  compelled  to  keep 
vibrating  for  a  long  time  and  be  then  left  to  vibrate  freely, 
the  vibrations  will  die  away  more  rapidly  than  they  would 
have  if  the  body  had  not  received  such  preliminary  treat- 
ment. Lord  Kelvin  found  that  the  torsional  vibrations 
of  a  wire  that  had  been  kept  in  torsional  vibration  for  a 
long  time  and  was  then  set  free  died  away  to  one-half  in 
44  or  45  vibrations,  while  the  vibrations  of  a  similar  wire, 
started  "  fresh,"  took  100  vibrations  to  fall  to  one-half. 
This  is  called  fatigue  of  elasticity. 

REFERENCES 

Tait's  "  Properties  of  Matter,"  Chapters  VIII  and  XL 
Poynting  and  Thomson's  "  Properties  of  Matter." 
Gray's  "  Treatise  on  Physics,"  Vol.  I,  Chapter  XL 
Article  on  "  Elasticity,"  Ency.  Brit. 
Johnson's  "  Materials  of  Construction." 


CHAPTER  XI 

MECHANICS   OF  FLUIDS 

138.  A   fluid   is   a   body  that   yields   to  the  smallest 
deforming  force ;  while  there  is  any  shearing  stress,  how- 
ever small,  in  a  fluid,  it  continues  to  flow;  that  is,  the 
amount  of  shear  continues  to  increase.     In  other  words, 
the  shear  modulus  of  a  fluid  is  zero. 

Fluids  are  divided  into  liquids  and  gases  according  to 
the  magnitude  of  their  bulk  moduli.  The  bulk  modu- 
lus of  a  liquid  is  large,  that  is,  it  takes  a  large  stress  to 
produce  a  small  strain  ;  when  the  pressure  on  water  is 
doubled,  its  volume  is  decreased  by  only  one  part  in  20,000. 
The  bulk  modulus  of  a  gas  is  small ;  when  the  pressure 
on  a  gas  is  doubled,  its  volume  is  decreased  by  one-half. 
The  difference  between  liquids  and  gases  seems  to  depend 
on  the  distances  between  particles,  the  particles  of  a  liquid 
being  comparatively  close  together  while  those  of  a  gas 
are  widely  separated. 

A  liquid  can  have  a  definite  "free  surface,"  that  is, 
a  surface  not  confined  by  a  solid  or  by  another  liquid ; 
whereas  a  gas  expands  so  as  to  occupy  the  largest  space 
open  to  it. 

139.  Direction  of  Force  on  the  Surface  of  a  Fluid.  —  When 
a  fluid  is  at  rest,  the  resultant  force  exerted  on  its  surface 

must  be  perpendicular  to  the  surface  ;  for  if  it  were  not, 

204 


FLUIDS  205 

it  would  have  a  component  parallel  to  the  surface,  and  this 
would  cause  a  flow  of  the  fluid.  Against  any  force  exerted 
on  it  a  fluid  exerts  an  equal  and  opposite  reaction  ;  hence 
a  fluid  at  rest  presses  perpendicularly  against  any  surface 
in  contact  with  it.  These  statements  apply  not  only  to 
the  contact  of  a  solid  and  a  liquid,  but  also  to  that  of  two 
liquids,  such  as  oil  and  water,  which  do  not  mix.  But 
when  a  fluid  is  in  motion,  the  force  between  it  and  the 
surface  of  a  body  in  contact  with  it  may  be  inclined  to 
the  surface  ;  a  fluid  flowing  through  a  pipe  tends  to  drag 
the  pipe  with  it. 

140.  Pressure  in  a  Fluid.  —  At  any  point  in  a  fluid  the 
part  of  the  fluid  on  one  side  of  an  imaginary  dividing 
plane  through  the  point  presses  against  the  fluid  on  the 
other  side.  If  the  fluid  on  one  side  of  the  plane  be  sup- 
posed removed  and  a  solid  surface  substituted,  the  latter 
will  sustain  the  pressure  of  the  remainder  of  the  fluid. 

The  whole  force  of  fluid  pressure  on  any  plane  surface  is 
called  the  thrust  (or  total  pressure)  on  the  surface,  and  the 
thrust  on  any  plane  surface  divided  by  the  area  of  the  sur- 
face is  called  the  average  pressure  on  the  surface.  The 
value  to  which  the  average  pressure  on  a  small  area  sur- 
rounding a  point  approaches  as  the  area  is  diminished  is 
called  the  pressure  (or  pressure  intensity)  at  that  point ; 
this  is  otherwise  expressed  as  the  thrust  per  unit  area  at 
the  point.  If  the  pressure  is  the  same  at  all  points  on  a 
surface,  it  equals  the  thrust  on  any  unit  of  area  of  the 
surface. 

Pressure  in  a  fluid  is  due  either  to  the  weight  of  the 
fluid,  as  in  the  case  of  water  in  a  tank,  or  to  force  applied 


206  MECHANICS 

to  some  part  of  the  surface  of  the  fluid,  e.g.  force  applied 
to  a  piston  that  closes  a  cylinder  containing  the  fluid.  (The 
effect  of  attractions  between  particles  of  the  fluid  need  not 
at  present  be  considered,  for  the  pressure  thus  caused  does 
not  act  on  a  body  immersed  in  or  exposed  to  the  fluid.) 

141.  Pressure  at  a  Point.  —  In  defining  the  pressure  at  a 
point  it  is  not  necessary  to  specify  any  direction,  for  at 
any  particular  point  the  pressure  has  the  same  magnitude 
in  all  directions.  For,  let  0  be  the  point  and  let  A10  and 
AZ0  be  any  two  directions  through  it. 
Around  0  describe  a  small  triangular 
prism,  two  sides  passing  through  ac  and 
ab  being  at  right  angles  to  A1 0  and  A2  0 
respectively,  while  a  third  side  passing 
through  be  is  equally  inclined  to  A^  0  and 
A20  and  the  ends  are  parallel  to  the  plane  of  A10 
and  A20.  The  fluid  within  the  prism  is  at  rest.  Hence 
the  whole  force  on  it  in  the  direction  be  is  zero.  But 
the  only  forces  in  this  direction  are  components,  in  the 
direction  be,  of  the  thrusts  P1  and  P2  on  the  sur- 
faces through  ab  and  ac  respectively.  Now  P1  and  P2 
are  equally  inclined  to  be,  and,  since  their  components  in 
the  direction  be  must  be  equal  and  opposite,  P1  and  P2 
must  also  be  equal  in  magnitude.  The  faces  through  ab 
and  ac  are  also  equal  in  area.  Hence  the  average  press- 
ures on  these  faces  must  be  equal  in  magnitude.  If  the 
dimensions  of  the  prism  be  supposed  diminished  without 
limit,  the  average  pressures  on  ab  and  ac  become  the  press- 
ure at  0  in  the  direction  A1 0  and  A2  0.  Hence  these  are 
also  equal.  But  Al  0  and  Az  0  are  any  directions  through 


FLUIDS 


207 


FIG.  79. 


0.     Hence  the  pressure  at  a  point  is  the  same  in  all  direc- 
tions. 

In  the  above  we  have  neglected  the  force  of  gravity  on  the  prism 
of  fluid ;  but  since  this  is  proportional  to  the  volume  of  the  fluid,  that 
is,  to  the  cube  of  the  dimensions  of  the  prism,  while  the  thrusts  on 
the  faces  are  proportional  to  the  squares  of  the  dimensions,  when  the 
prism  is  reduced  without  limit  the  force  of  gravity  vanishes  in  com- 
parison with  the  thrusts. 

142.  Pressure  at  Different  Points  in  a  Fluid.  —  (1)  Let  A 
and  B  be  two  points  in  a  horizontal  line,  and  let  AB  be 
wholly  in  the  fluid.  Around  AB  as 
axis  describe  a  cylinder  with  vertical 
ends.  The  fluid  within  the  cylinder 
is  at  rest.  Hence  the  resultant  hori- 
zontal force  on  it  is  zero.  Since  its 
weight  acts  vertically  and  the  thrusts 
on  its  sides  are  perpendicular  to  AB,  the  only  forces  in 
the  direction  AB  are  the  thrusts  on  its  ends,  and  these 
must  therefore  be  equal  in  magnitude  and  opposite  in 
direction.  Hence  the  pressure  per  unit  area  at  A  must 
equal  that  at  B.  Thus  the  pressure  is  the 
same  at  all  points  in  the  same  horizontal  plane 
in  the  fluid. 

(2)  Let  A  and  B  be  two  points  in  the  same 
vertical  line  AB  in  the  fluid.  Around  AB  as 
axis  describe  a  cylinder  with  horizontal  ends. 
The  fluid  within  the  cylinder  is  at  rest  and  the 
resultant  vertical  force  on  it  is  therefore  zero. 
Therefore,  if  the  thrust  on  the  end  A  be  Pl 
and  that  on  the  end  B  be  Pv  and  if  A  be  above  B,  P^—P\ 
must  equal  the  weight  of  the  cylinder.  If  the  density  of 


FIG. 


208  MECHANICS 

the  fluid  (or  its  mass  per  unit  volume)  be  />,  and  if  the 
length  of  the  cylinder  be  h  and  its  cross-section  be  a,  the 
volume  of  the  cylinder  is  ha,  its  mass  is  hap,  and  its  weight 

is  hapg. 

.'.  PZ-P1  =  hapg 

P       P 

and  — =  hog. 

a        a 

The  pressure  on  the  end  A  is  a  uniform  pressure  and 
therefore  the  pressure,  pv  at  A  equals  P1  -*-  a.  Similarly, 
the  pressure,  p2,  at  B  equals  _P2  -+-  a. 

Hence  p%  —  pl  =  hpg. 

(3)  Any  two  points  A  and  B  in  the  fluid  can  be  con- 
nected by  a  broken  line  ACD  •••  B  consisting  of  horizontal 
and  vertical   steps.       Along   each   horizontal 
step  there  will  be  no  change  of  pressure  and 
the  total  change  of  pressure  along  the  vertical 
steps  will  be  hpg,  h  being   the   difference  of 
level  of  A  and  B. 
FIG.  81.  in  the  above  we  have  assumed  that  the  den- 

sity p  is  the  same  at  all  points  in  the  fluid.  This  is 
practically  true  for  small  bodies  of  fluid,  but  it  is  far  from 
true  for  large  bodies  such  as  the  atmosphere  and  the  ocean. 
The  density  of  a  gas  is  so  small  that,  unless  h  be  very 
large,  hpg  is  very  small  compared  with  pl  or  p2.  Hence 
in  moderate  volumes  of  a  gas  the  pressure  may  be  con- 
sidered as  everywhere  the  same. 

143.  Surface  of  Contact  of  Two  Fluids.  —  The  surface  of 
contact  of  two  fluids  of  different  densities  which  are  at 
rest  and  do  not  mix  is  horizontal.  For,  take  two  points 


FLUIDS  209 

P  and   Q  on  the   surface   of  contact,  and  let  a  vertical 
through  P  meet  a  horizontal  through  Q  in  a  point  A  in 

the  fluid  of  density  /o,  while  a  vertical    

through   Q  meets  a  horizontal  through 
P  at  the  point  B  in  the  fluid  of  den- 
sity //.     The  pressures  at  A  and  Q  are 
equal,  and  the  pressures  at  P  and  B  are 
equal.     Hence  the  increase  of  pressure  from  A  to  P  equals 
that  from  Q  to  B,  or,  denoting  the  common  length  of  AP 
and  QB  by    A,    hpg  =  hp'g,    and   therefore   h(p—p')=Q. 
Hence,  since  p  and  /o'  are  unequal,  h  must  be  zero,  or  P. 
and  $  must  be  in  a  horizontal  line. 

It  follows  from  the  above  that  the  free  surface  of  a 
liquid  at  rest  is  horizontal.  This  is  also  readily  seen  by 
considering  that,  if  the  surface  were  not  horizontal,  the  ver- 
tical force  of  gravity  would  have  a  component  parallel  to  the 
surface,  and  this  would  cause  motion  parallel  to  the  surface. 

If  the  pressure  on  the  horizontal  free  surface  of  a  liquid 
is  P,  the  pressure  p  at  a  depth  h  below  the  surface  is 
greater  than  P  by  gph,  or  p  =  P  +  gph, 

144.  Transmissibility  of  Fluid  Pressure  (Pascal's  Principle) . 
—  In  a  fluid  of  constant  density  and  at  rest,  the  difference  of 
pressure  between  two  points  depends  only  on  the  difference 
of  level  of  the  points  and  the  density.  Hence,  an  increase 
of  pressure  at  any  point  is  accompanied  by  an  equal 
increase  at  all  points.  This  is  known  as  Pascal's  principle 
(first  stated  by  Pascal  in  1653).  It  will  be  noticed  that 
it  applies  strictly  only  to  a  fluid  of  constant  density,  that 
is,  an  incompressible  fluid.  If  an  increase  of  pressure 
affected  the  density  to  an  appreciable  extent,  it  would 


210  MECHANICS 

cause  a  change  in  the  difference  of  pressure  between  two 
points  not  at  the  same  level.  Liquids  are  so  nearly 
incompressible  that  the  principle  is  practically  true  for  all 
liquids.  Gases  are  more  compressible,  but,  on  account  of 
their  small  density,  the  pressure  is  practically  the  same  at 
all  points,  provided  the  volume  be  not  enormously  great. 
Hence  the  principle  is  also  practically  true  for  gases. 

In  the  hydraulic  press  a  small  cylinder  containing  a 
piston  is  in  communication  with  a  large  cylinder  con- 
taining a  correspondingly  large 
piston,  both  cylinders  being  filled 
with  a  liquid.  If  the  pressure  in 
the  liquid  be  p  and  the  areas  of 


the  pistons  be  s  and  S  respectively, 

the  thrust  on  the  small  piston  will  be  ps  and  that  on  the 
large  piston  pS.  A  force  ps  applied  to  the  rod  of  the  small 
piston  will  produce  a  force  pS  on  the  rod  of  the  large 
piston.  (In  the  forging  press  of  the  Bethlehem  Steel 
Works  a  force  of  14,000  tons  is  thus  produced  by  a  water 
pressure  of  8000  Ibs.  per  square  inch.) 

145.    Thrust  on  a  Plane  Surface  immersed  in  a  Liquid.  —  Let 

the  area  of  a  plane  surface  immersed  in  a  liquid  be  A,  and 
suppose  A  divided  up  into  a  large  number  of  parts  av  «2,  ••• 
each  so  small  that  the  pressure  over  it  may  be  regarded  as 
uniform.  If  liv  A2,  •••  are  the  respective  depths  of  these 
parts,  and  if  P  is  the  pressure  on  the  surface  of  the  liquid, 
and  F  the  thrust  on  the  immersed  surface, 

F=  (P  +ffph1)a1  +  (P  +  <7^2>2  +  - 
=  POi  +  a2  -f  ...)  +gp(h 
=  PA  +ffp(h1a1  +  V2  + 


FLUIDS  211 

If  the  depth  of  the  centroid  of  the  surface  (i.e.  the 
centre  of  mass  of  a  thin  uniform  disk  having  the  shape 
of  the  surface)  is  H  (§  79), 


Hence  the  thrust  on  a  plane  surface  is  the  same  as  if  the 
surface  were  horizontal  and  at  the  depth  of  its  centroid. 

146.  Archimedes'  Principle.  —  The  resultant  force  which  a 
fluid  at  rest  exerts  on  a  body  immersed  in  it  equals  the 
weight  of  the  fluid  displaced  and  acts  vertically  upward 
through  the  centre  of  gravity  of  the  fluid  before  it  was  dis- 
placed. For,  call  the  immersed  body  B  and  the  fluid  dis- 
placed S.  Before  it  was  displaced  S  was  at  rest  and  it 
must,  therefore,  have  been  sustained  by  a  force  equal  to  its 
weight  and  acting  upward  through  its  centre  of  gravity. 
When  B  is  introduced  (the  level  of  the  liquid  being  kept 
the  same),  the  pressure  on  any  part  of  its  surface  is  the 
same  as  the  pressure  that  acted  on  the  corresponding  part 
of  the  surface  of  S.  Hence  B  must  be  buoyed  up  by  a 
force  equal  to  the  weight  of  S  acting  through  the  centre 
of  gravity  of  S.  The  centre  of  gravity  of  the  displaced 
fluid  is  called  the  centre  of  buoyancy  of  the  body  immersed. 
When  the  immersed  body  is  homogeneous,  its  centre  of 
buoyancy  coincides  with  its  centre  of  gravity. 

If  the  volume  of  the  body  immersed  is  v  and  its  density 
(supposed  uniform)  is  p,  its  weight  is  vpg.  If  pf  is  the 
density  of  the  fluid,  the  weight  of  the  fluid  displaced  is 
vp'g  and  this  is,  therefore,  the  apparent  loss  of  weight  of 
the  immersed  body.  Hence  the  ratio  of  the  weight  of  the 


212 


MECHANICS 


body  to  its  apparent  loss  of  weight  when  immersed  is  p  :  pr . 
Thus  by  weighing  a  body  in  air  and  in  a  liquid,  the  den- 
sity of  the  body  can  be  found  if  that  of  the  liquid  be 
known,  or  the  density  of  the  liquid  can  be  found  if  that  of 
the  solid  be  known. 

147.  Specific  Gravity  and  Density.  —  The  -specific  gravity 
of    any  substance  is  the  ratio  of   its  density  to  that  of 
water  at  4°  C.,  or  the  ratio  of  the  mass  or  weight  of  any 
volume  of   the  substance  to  that  of  an  equal  volume  of 
water  at  4°  C.     Since  the  mass  of  a  c.c.  of  water  at  4°  C. 
may  be  taken  as  1  g.  (§  53),  the  density  of  water  at  4°  C.  is 
1,  and  therefore  in  the  C.  G.  S.  system  the  specific  gravity  of 
a  substance  is  the  same  as  its  density.     In  the  F.  P.  S. 
system  the  density  of  water  is  62.4  (Ibs.  per  cu.  ft.),  and 
the  density  of    any  substance  equals  its  specific  gravity 
multiplied  by  62.4. 

148.  Hydrometers.  —  The  common  hydrometer  (or  hy- 
drometer of  variable  immersion)  is  an  instrument  for  find- 
ing the  specific  gravities  of  liquids. 
It  is  made  of  glass  and  consists  of  a 
body,  in  the  form  of  two  bulbs,  with  a 
tube   or  stem  attached.     The  lower 
bulb  is  weighted   with   mercury   so 
that  the  instrument  will  float  stably 
with  the  stem  vertical.      When  in  a 
liquid  it  sinks  to  a  depth  that  indi- 
cates, by  a   scale    on    the   tube,    the 
specific   gravity  of   the   liquid.     To 
construct   a   suitable   scale    for    any 

hydrometer  the  depth  is  noted  to  which  it  sinks  (1)  in 


FIG.  84. 


FLUIDS  213 

water,  (2)  in  a  liquid  of  known  specific  gravity  sr 
This  gives  the  water  mark  and  the  mark  on  the  scale 
for  a  density  sr  Let  the  distance  between  them  be 
d  and  let  the  distance  from  the  water  mark  to  the 
place  where  the  s  mark  should  be  put  be  x.  If  v  is 
the  volume  the  instrument  displaces  when  floating  in 
water,  the  volume  it  displaces  when  floating  in  liquids 
of  specific  gravities  sx  and  s  respectively  must  be 

-  and  -  respectively.     Hence,  if  a  is  the  area  of  cross- 

Si  S 

section  of  the  stem  and  if  s  and  s1  are  both  greater  than  1, 


V 

v  —  =  ax, 
s 


whence  — : 

a 

s1 

(If  Sj  and  s  be  less  than  1,  the  minus  signs  must  be 
replaced  by  plus  signs.) 

Exercise  XXXIV.    Archimedes'  Principle 

(1)  To  the  inner  wall  of  a  straight  glass  tube  (a  shade  for  a 
Welsbach  burner)  a  millimetre  scale  on  thin  paper  is  fastened,  so  that 
the  scale  is  parallel  to  the  axis  of  the  tube  and  the  paper  reaches  to 
about  2  cm.  of  one  end  of  the  tube.  This  end  is  closed  by  a  thin 
cork  pushed  a  small  distance  into  the  tube  and  covered  by  a  layer  of 
paraffin  wax  that  just  fills  the  end  of  the  tube. 

The  tube  is  then  floated  in  a  jar  of  water  and  disks  of  lead  slightly 
smaller  in  diameter  than  the  tube  (or  lead  shot)  are  dropped  in  until 
the  tube  will  float  stably.  The  depth  of  the  tube  is  read  on  the  scale 
by  glancing  along  the  under  surface  of  the  water  in  the  jar.  The 


214  MECHANICS 

tube  is  then  removed,  dried,  and  weighed  with  its  contents.  The 
volume  of  the  tube  immersed  is  calculated  and  compared  with  the 
weight.  These  operations  should  be  repeated  with  different  depths 
of  immersion. 

The  same  observation  should  be  repeated  with  some  other  liquid, 
such  as  a  strong  brine,  instead  of  water,  and  the  density  of  the  brine 
calculated. 

(2)  Find  the  specific  gravity  of  aluminium  (or  other  metal)  by 
weighing  a  block  of  the  metal,  first  in  air  and  then  in  water,  by  means 
of  a  calibrated  spring  and  a  mirror  scale.     Then  find  the  specific 
gravity  of  the  brine  used  in  (1)  by  weighing  the  block  in  the  brine. 
Air -bubbles  clinging  to  the  block  may  be  removed  by  a  bent  wire. 

(3)  Construct  a  scale  for  a  hydrometer.     First  weight  the  instru- 
ment with  shot  so  that  it  will  float  in  water  with  the  stem  nearly 
immersed.     Slip  a  paper  millimetre  scale  (a  strip  of  cross-section 
paper  will  do)  into  the  stem  (which  is  closed  by  a  cork)  and  note  the 
reading  in  water  and  in  the  brine  used  in  (1)  and  (2).     This  gives 
d  (§    148).     Then  calculate  the   values  of  x  for   specific  gravities 
increasing  by  .05  and  lay  them  off  on  a  strip  of  paper  precisely  similar 
to  the  millimetre  scale.     Having  placed  this  in  the  instrument,  test  it 
in  water  and  in  the  brine.     Then  make  two  mixtures,  the  first  con- 
sisting of  two  parts  (by  volume)  of  brine  and  one  part  of  water, 
the  second  with   these  proportions  reversed.     Calculate  the  specific 
gravities  of  these  and  also  find  them  by  means  of  the  hydrometer. 

DISCUSSION 

(a)  Why  are  the  divisions  of  a  hydrometer  not  equally  spaced  ? 
(&)  The  density  of  ice  is  .92.     What  part  of  the  volume  of  an 
iceberg  is  under  salt  water  of  density  1.026? 

(c)  Archimedes  weighed  the  crown  of  Hiero  in  water  and  found  a 
decrease  of  T\  of  its  weight,  while  a  block  of  gold  and  a  block  of  silver, 
each  of  the  same  weight  in  air  as  the  crown,  lost  ,.in  water  74r  and  ^ 
respectively  of  the  common  weight.     Of  what  did  the  crown  consist? 

(d)  A  block  of  metal  of  specific  gravity  9.3  floats  partly  in  oil  of 
specific   gravity  .9  and  partly  in  mercury  of   specific  gravity  13.5. 
What  part  of  its  volume  is  in  each? 


FLUIDS 


215 


FIG.  85. 


(e)  Explain  as  fully  as  possible  the  difficulty  found  in  getting  the 
tube  in  (1)  to  float  stably  when  not  sufficiently  weighted. 

149.  Equilibrium  of  Floating   Bodies.  —  Two  forces  act 
on  a  floating  body:  (1)  the  weight  of  the  body  acting  at 
the  centre  of  gravity  6r  of  the  body; 

(2)  the  resultant  upward  pressure  of 
the  liquid  acting  at  the  centre  of 
gravity  0  of  the  liquid  displaced. 
When  the  body  is  at  rest,  G-  and  0 
must  be  in  the  same  vertical  line.  Let 
AB  be  the  line  in  the  body  which  con- 
tains Gr  and  C  when  the  body  is  at  rest. 
If  the  body  be  slightly  displaced,  the 
centre  of  gravity  of  the  liquid  displaced  will  be  at  some 
point  C1 .  We  shall  consider  only  the  case  in  which  6r,  (7, 
and  0'  lie  in  a  vertical  plane.  The 
point  M  in  which  a  vertical  line 
through  0'  cuts  AB  is  called  the  meta- 
centre  of  the  body.  K  is  evident  that 
if  M  be  above  Gr  (Fig.  85),  the  couple 
acting  on  the  body  will  tend  to  right 
it,  and  the  equilibrium  will  be  stable ; 
but  if  M  be  below  G-  (Fig.  86),  the 
equilibrium  will  be  unstable.  The 
two  cases  are  illustrated  by  a  rod  or  a  long  cylinder 
(1)  on  its  side,  (2)  on  end,  in  water.  A  ship  has  two 
metacentres,  —  one  for  rolling  and  one  for  pitching 
motion. 

150.  Flow  of  Liquid  from  an  Orifice.  —  Liquid  flows  from 
an  orifice  in  a  vessel  because  the  pressure  in  the  liquid  is 


FIG.  86. 


216  MECHANICS 

greater  than  that  in  the  air.  The  escaping  liquid  gains 
kinetic  energy,  while  the  whole  body  of  liquid  in  the 
vessel  falls  to  a  lower  level  and  so  loses 
potential  energy.  Suppose  the  orifice  to  be 
opened  long  enough  for  a  small  mass  m  to 
escape.  If  its  velocity  be  v,  its  kinetic 
energy  will  be  ^  mv2.  The  state  of  the  liquid 
in  the  vessel  is  the  same  as  if  the  mass  m  had 

been  removed  from  the  surface  and  lowered 
FIG.  87. 

to  the  orifice.  Thus  the  decrease  of  potential 
energy  is  mgh,  where  h  is  the  depth  of  the  orifice  below 
the  surface.  Hence 

1  mv2  =  mgh 

or  v  =  V2  gh. 

This  is  also  the  velocity  attained  by  a  body  in  falling 
freely  a  distance  h ;  this  is  called  Torricelli's  Law.  If  the 
escaping  jet  were  turned  vertically  upward,  it  would  rise 
to  the  level  of  the  surface,  if  friction  could  be  neglected. 

The  quantity  of  liquid  that  escapes  in  any  time  cannot 
be  calculated  from  the  value  of  v  and  the  area  of  the 
orifice,  owing  to  the  fact  that  just  outside  of  the  orifice 
the  jet  contracts  somewhat.  If,  however,  a  is  the  area 
of  the  smallest  cross-section  of  the  jet  (called  the  vena 
contracta),  the  volume  that  escapes  in  time  t  is  vat.  The 
ratio  of  a  to  the  area  of  the  orifice  depends  on  the  form  of 
the  orifice  and  the  velocity  of  escape,  and  it  may  be  altered 
by  the  insertion  of  a  tube  (or  ajutage")  in  the  orifice. 

Exercise  XXXV.    Flow  of  Liquid  from  an  Orifice 

A  tin  tank  (12"  x  4"  x  2")  is  mounted  at  the  upper  left-hand 
corner  of  a  cross-section  board  as  in  Exercise  III.  The  tank  is  filled 


FLUIDS  217 

nearly  to  the  brim,  and  an  aperture  in  the  tank  is  opened  by  pressing 
a  lever  on  the  side  of  the  tank.  The  out-flowing  liquid  is  caught  in 
another  tank  attached  to  the  board.  The  parabola  of  descent  is  ob- 
tained by  slightly  turning  the  tank,  so  that  the  water  leaves  a  streak 
on  the  board,  or  by  the  method  of  Exercise  III. 

From  the  parabola  the  velocity  of  the  escaping  liquid  is  found  as 
in  Exercise  III.  For  this  calculation  the  values  of  x2  -4-  y  obtained 
from  points  within  a  foot  of  the  tank  are  to  be  preferred,  since  the 
stream  breaks  up,  and,  owing  to  impacts  and  air  friction,  the  paths 
of  the  particles  do  not  continue  to  be  true  parabolas.  The  experi- 
mental values  of  the  velocity  will  in  all  cases  be  somewhat  less  than 
the  velocity  calculated  from  Torricelli's  Law,  and,  as  the  difference 
will  vary  with  the  depth  of  the  water  in  the  tank,  several  determina- 
tions should  be  made.  In  making  each  determination  the  aperture 
should  be  open  for  as  short  a  time  as  possible  so  that  the  level  of  the 
water  in  the  tank  may  not  change  appreciably. 

DISCUSSION 

(a)  Explain  the  difference  between  the  experimental  results  and 
the  values  given  by  Torricelli's  Law. 

(6)  What  additional  force  acts  on  the  tank  when  the  orifice  is 
opened  ? 

(c)  How  high  would  the  jet  rise  if  it  issued  in  some  oblique 
direction  ? 

(d)  Calculate  the  kinetic  energy  of  the  liquid  in  the  tank  during 
the  outflow. 

(e)  Why  does  the  jet  break  up  at  a  distance  from  the  orifice  ? 
(/)  What  would  be  the  result  of  inserting  an  outflow  tube  to 

get  a  more  definite  stream? 

151.  Flow  past  an  Obstruction.  —  When  a  stream  meets 
an  obstacle,  the  particles  of  the  fluid  are  deflected  from 
straight  lines  and  travel  past  the  obstacle  in  curves.  The 
obstacle  suffers  a  pressure  in  giving  curvature  to  the  paths 
of  the  particles  just  as  the  outer  rail  of  a  curved  track 
suffers  pressure  in  curving  the  path  of  a  train.  If  the 


218  MECHANICS 

obstacle  be  a  vertical  disk  or  board  symmetrical  about  a 
vertical  axis,  the  pressures  on  opposite  sides  of  the  vertical 
axis  will  be  equal  provided  the  disk  be  at  right  angles 
to  the  stream.  If  it  be  inclined  to  the  stream,  the  "  up 
stream  "  side  will  deflect  the  fluid,  which  will  flow  down 
along  the  disk,  and  there  will  therefore  be  a  moment  of 
force  tending  to  set  the  disk  across  the  stream. 
The  effect  may  be  illustrated  by  sweeping 
through  the  air  a  frame  covered  with  paper  and 
free  to  rotate  about  an  axis,  as  illustrated  in 
Fig.  88.  The  stability  of  a  kite  depends  partly 
IG'  '  on  the  same  principle.  (Additional  stability  is 
given  to  a  Japanese  (or  tailless)  kite  by  curving  the 
horizontal  rib  backward.  It  is  readily  seen  from  the 
figure  that  a  tilt  from  the  normal  position 
will  greatly  increase  the  pressure  on  the  for-  Hi/'' 

ward  side  and  decrease  that  on  the  other.)  ^Jl 

A  similar  effect  takes  place  when  a  disk  is 
moved  through  a  fluid  at  rest.     Thus  a  sheet 
of  paper  or  a  leaf,  falling  through  air,  tends  to  become  hori- 
zontal, and  the  same  is  true  of  a  coin  sinking  through  water. 

152.   Speed  and  Pressure.  —  If  the  cross-section  of  a  tube 
in  which  liquid  flows  steadily  is  not  uniform,  the  speed  of 

the  particles  of  the  liquid  must 
increase  as  they  come  to  a  narrow 
part  of  the  tube,  since  the  same 
amount  of  liquid  passes  through 
all  cross-sections.  Hence  there 
must  be  a  resultant  forward  force 
acting  on  the  liquid,  or  the  pressure  behind  in  the  wider 


FLUIDS  219 

cross-section  must  be  greater  than  that  ahead  in  the  con- 
traction. Thus  the  pressure  decreases  as  the  speed  in- 
creases, and  conversely.  For  a  similar  reason,  when  air  is 
forced  out  between  two  plates,  the  pressure 
between  the  plates  is  less  than  that  outside  and 
the  plates  are  pressed  together.  (The  appa- 
ratus sketched  in  Fig.  91,  consisting  of  two 
corks  and  a  glass  tube  through  which  air  is 
blown,  will  illustrate  this.)  The  same  prin- 
ciple is  applied  in  the  atomizer,  the  steam  injector,  the 
ball  nozzle,  etc.  From  the  change  of  pressure  in  a  liquid 
as  it  passes  through  a  throat  in  a  tube  the  speed  can  be 
deduced;  this  is  the  method  used  in  the  Venturi  metre 
for  gauging  flow  of  water. 

The  "  curve  "  of  a  rotating  tennis-ball  or  baseball  is  due  to  the 
same  cause.     Suppose,  for  simplicity,  that  the  ball  is  not  moving 
forward,  but  is  rotating  about  a  vertical  axis,  and  that  a  current  of  air 
is  blowing  horizontally  toward  it.      The  rotating  ball  is  carrying  a 
whirl  of  air  around  with  it.    On  one  side,  A,  the  effect 
of  the  rotation  of  the  ball  is  to  cause  a  decrease  in 
the  velocity  of  the  air  blowing  past  the  ball,  while  on 
the  other  side,  B,  it  causes  an  increase.     Hence  the 
pressure  at  A  is  greater  than  that  at  B,  and  there  is, 
therefore,  a  force  acting  on  the  ball  in  the  direction 
FIG  92  AB>      If    now    the  rotating  ball  has  a  motion  of 

translation  in  air  otherwise  at  rest,  the  effect  will 
be  the  same  and  the  ball  will  have  an  acceleration  in  the  direction 
AB  and  will,  therefore,  move  in  a  path  curved  to  the  left  in  the  case 
represented  in  the  figure.  (The  "  curve  "  of  a  ball  may  be  illustrated 
by  "  serving  "  a  toy  balloon  with  a  "  cut "  from  the  hand.) 

153.   Viscosity.  —  A  body  of  fluid  continues  to  change  in 
form  so  long  as  there  is  the  smallest  shearing  stress  acting 


220  MECHANICS 

on  it ;  but  the  rate  of  change  of  shape  for  a  given  shearing 
stress  is  different  for  different  fluids.  For  example,  a 
liquid  flows  down  an  inclined  plane,  however  slight  the 
inclination,  but  the  rate  of  flow  is  different  for  different 
liquids;  the  shearing  stress  is,  in  this  case,  due  to  the 
component  of  gravity  down  the  plane.  Careful  experi- 
ments have  shown  that  in  all  cases  the  rate  of  change 
of  shape  or  rate  of  shearing,  after  it  has  become  steady,  is 
accurately  proportional  to  the  magnitude  of  the  shearing 
stress.  Hence  the  internal  frictional  resistance,  which 
just  counterbalances  the  external  force  and  so  prevents 
acceleration,  must  also  be  proportional  to  the  rate  of 
shearing.  The  internal  resistance  is  called  the  viscosity 
of  the  fluid  and  the  ratio  of  the  shearing  stress  to  the  rate 
of  shearing  is  called  the  coefficient  of  viscosity  of  the  fluid. 
For  clearness  this  definition  of  the  coefficient  of  vis- 
cosity may  be  interpreted  as  follows :  Suppose  the  space 
between  two  large  plates  A  and  B  to 
be  filled  by  a  fluid.  Let  B  be  kept  at 
rest  while  A  is  kept  moving  parallel 
to  B  with  a  steady  velocity  v.  In  a 
short  time  £,  A  travels  a  distance  vt, 
and  if  the  distance  between  the  plates  is  d,  the  shear 
produced  is  vt  -r-  d.  Hence  the  rate  of  shearing  is  v  -4-  d. 
If  the  area  of  each  plate  is  a  and  the  force  applied  to  each 
plate  is  F,  the  measure  of  the  shearing  stress  is  F+a. 
Hence,  denoting  the  coefficient  of  viscosity  by  /-t, 

^F^v^Fd 

ad      av' 

and  therefore  F  =  M—  • 

d 


FLUIDS 


221 


This  equation  is  frequently  taken  as  the  definition  of  /-t, 
the  other  letters  having  the  meanings  already  assigned  to 
them.  Taking  the  case  in  which  a,  v,  and  d  are  all  unity, 
we  get  the  following  definition  of  p :  the  coefficient  of 
viscosity  of  a  viscous  material  is  the  tangential  force  on 
unit  of  area  of  either  of  two  horizontal  planes  at  the  unit 
distance  apart,  one  of  ivhich  is  fixed  while  the  other  moves 
with  the  unit  of  velocity,  the  space  between  them  being  filled 
by  the  viscous  material  (Maxwell). 


Exercise  XXXVI.    Viscosity 

A  vertical  steel  rod  (1.5  cm.  in  diameter)  is  held  between  needle 
points,  one  of  which  is  in  a  horizontal  bar  (clamped  to  uprights) 
while  the  other  is  in  a  plug  that  closes  the  lower  end  of  a  brass  tube 
(about  1.7  cm.  in  internal  di- 
ameter) containing  glycerine. 
The  tube  rests  on  the  table 
and  its  upper  end  is  steadied 
by  an  adjustable  clamp  at- 
tached to  a  second  horizontal 
bar.   A  cord  that  passes  over 
a  pulley  and  carries  a  scale 
pan  is  wrapped  around  the 
rod.     (The  pulley  should  be 
kept  well  oiled.) 

Weights  are  placed  in  the 
pan  and  the  time  required 
for  the  pan  to  descend  from  _ 
a  definite  level  to  the  floor  is 
observed  several  times  with 
care,  the  pan  being  released 

exactly  on  a  tick  of  the  clock.  The  velocity  of  rotation  of  the  rod 
becomes  constant  almost  immediately  after  the  release  of  the  pan 
(it  is  just  perceptible  that  there  is  a  momentary  acceleration  which 


FIG.  94. 


222  AfECHANICS 

produces  a  velocity  too  great  to  be  maintained  steadily  and  that  the 
velocity  falls  at  once  to  the  constant  value).  From  the  distance  of 
descent  and  the  time  the  velocity  is  calculated.  The  observations 
should  be  repeated  with  various  weights  in  the  pan. 

The  various  velocities  should  then  be  plotted  as  abscissae  with 
weights  as  ordinates.  The  result  should  be  a  very  satisfactory  straight 
line.  This  line  will  cut  the  vertical  axis  at  a  distance  from  the  origin 
that  represents  the  friction  of  the  bearings  and  pulley.  Subtracting 
this  friction  from  all  the  ordinates,  a  line  through  the  origin  showing 
the  constancy  of  the  ratio  of  shearing  stress  to  rate  of  shear  will  be 
obtained. 

The  friction  of  the  bearings  and  pulley  may  be  found  directly  by 
removing  the  glycerine  and  finding  what  weight  attached  to  the  cord 
will  keep  the  axis  in  steady  rotation  (the  lower  bearing  should  be  lubri- 
cated with  glycerine  as  in  the  preceding) . 

To  find  the  coefficient  of  viscosity  of  the  glycerine  the  internal  diam- 
eter of  the  tube  must  be  known.  It  may  be  determined  by  weighing 
the  tube  (and  plug)  empty  and  then  when  filled  with  water.  The 
value  of  a  (§  153)  may  be  calculated  from  the  mean  of  the  diameter 
of  the  rod  and  the  internal  diameter  of  the  tube.  The  ratio  of  F  to  v 
is  taken  from  the  line  through  the  origin. 

(The  plug  may  be  made  so  as  to  screw  in,  but  this  is  not  essential ; 
a  well-turned  brass  plug  that  can  be  forced  in  is  sufficient.  For  the 
lower  bearing  use  a  large,  thick-pointed  needle  forced  into  a  hole 
drilled  through  the  plug.) 

DISCUSSION 

(a)  Meaning  and  definition  of  viscosity. 

(b)  Why  does  the  weight  not  fall  with  an  acceleration  as  if  the 
resistance  were  ordinary  friction? 

(c)  Why  should  the  velocity  become  for  a  moment  too  great  to  be 
maintained? 

(d)  In  what  way  would  the  motion  differ  if  the  rod  carried  a  disk 
of  considerable  moment  of  inertia? 

(e)  What  becomes  of  the  potential  energy  of  the  descending  weight  ? 
(/)  Why  does  a  body  (e.g.  a  raindrop  or  a  parachutist)  falling  a 

long  distance  through  the  air  attain  a  steady  velocity  ? 


FLUIDS  223 

154.  Flow  through  a  Capillary  Tube.  —  When  a  fluid  flows 
without  eddies  through  a  capillary  tube  (that  is,  a  tube  of 
very  small  bore),  each  particle  moves  in  the  direction  of 
the  length  of  the  tube.  All  particles  in  a  cylindrical  layer 
move  with  the  same  velocity.  Hence  the  flow  consists  in 
a  sliding  of  layer  over  layer.  There  is  very  complete 
evidence  that  the  fluid  in  contact  with  the  surface  of  the 
tube  does  not  slip  on  the  solid,  but  adheres  to  it.  Assum- 
ing this,  it  can  be  shown  that,  if  r  is  the  radius  of  the  tube 
and  I  its  length,  and  if  the  difference  of  pressure  in  the 
fluid  at  the  ends  of  the  tube  is  p,  the  volume  that  flows 
out  of  the  tube  in  unit  time  is 


Numerous  experiments  have  verified  this  formula  for  tubes 
of  different  lengths  and  radii  and  for  different  pressures. 
This  shows  that  there  is  no  slipping  of  the  liquid  on  the 
surface  of  the  tube,  for  such  slipping  would  allow  an  out- 
flow not  included  in  the  formula. 

From  the  rate  of  outflow  of  a  fluid  through  a  tube  of 
measured  dimensions,  the  value  of  JJL  for  the  fluid  can  be 
deduced,  and  this  is  the  most  common  method  of  measur- 
ing the  coefficient  of  viscosity  of  a  fluid. 

SURFACE  TENSION  AND  CAPILLARITY 

155.  Intermolecular  Forces.  —  Many  facts  show  that  par- 
ticles of  any  form  of  matter  attract  one  another  with  very 
great  forces  when  they  are  very  close  together,  but  these 
forces  decrease  so  rapidly  with  distance  that  they  become 
negligible  beyond  a  certain  distance  called  the  range  of 


224  MECHANICS 

molecular  forces.  Fracture  of  a  brittle  body  consists  in 
slightly  separating  the  particles  of  the  body  so  that  they 
cease  to  attract.  Two  metallic  surfaces  may  be  brought 
quite  close  together  without  showing  any  attraction,  but  if 
brought  very  close  by  pressure  or  welding,  they  will  adhere. 
A  sheet  of  glass  brought  toward  a  water  surface  is  not 
attracted  sensibly  until  it  touches  the  water,  when  the 
latter  adheres  to  it.  If  water,  carefully  freed  from  air, 
be  heated  so  as  to  fill  a  tube  with  a  very  fine  stem  which 
is  then  sealed  off,  the  water  will  sometimes  continue  to 
cling  to  and  fill  the  tube  in  spite  of  its  tendency  to  con- 
tract as  it  cools. 

The  range  of  molecular  forces  is  not  yet  well  ascertained, 
but  there  is  reason  to  believe  that  it  is  about  .00005  mm. 
A  sphere  of  this  radius  is  called  the  sphere  of  influence  of 
the  particles. 

156.  Surface  Layer  of  a  Liquid.  —  A  particle  p  in  the 
liquid  at  a  greater  distance  from  the  surface  than  the 
range  of  molecular  forces  r  is  equally  attracted  in  all 
directions,  and  the  resultant  force  of  attraction  on  it  is 

zero.  But  a  particle  pl  at  a 
distance  less  than  r  from  the 
surface  is  more  attracted 
inward  than  outward.  For, 
with  p1  as  centre,  describe  a 

FIG.  95.  ,  .  ,  , .  —., 

sphere  with  r  as  radius.    That 

part,  abc,  of  the  sphere  which  is  out  of  the  liquid  contains 
only  a  relatively  very  small  number  of  particles  of  air  or 
vapor.  Let  ed  be  a  plane  through  pl  parallel  to  the  sur- 
face, and  let  fg  be  another  plane  parallel  to  the  surface 


FLUIDS  225 

and  cutting  off  a  segment  fgh  equal  to  abc.  The  attrac- 
tions on  p1  of  the  particles  in  acde  and  defg  will  neutralize 
one  another,  and  the  attractions  of  the  particles  in  fyh  will 
constitute  an  unbalanced  force  inward  acting  on  the  par- 
ticle pv  A  particle,  such  as  jt?2,  in  the  surface  will  be 
attracted  inward  by  the  resultant  of  the  attractions  of  all 
the  particles  in  a  hemisphere  of  the  sphere  of  influence. 

157.  Tendency  of  Surface  to  Contract.  —  The  effect  of  the 
inward  attractions  on  the  particles  near  the  surface  of  a 
liquid  is  a  tendency  of  the  surface  to  contract  to  the  form 
of  smallest  area  possible  under  the  circumstances.     For  a 
given  volume  the  form  of  smallest  surface  is  a  sphere,  and 
this  is  accordingly  the  form  that  a  body  of  liquid  assumes 
when  other  forces,  such  as  gravity,  do  not  interfere.     For 
example,  a  small  drop  of  mercury  on  a  glass  plate  is  prac- 
tically spherical;  its  weight  and  the  pressure  of  the  plate 
are  too  slight  to  cause  any  appreciable  flattening.    A  drop 
of  any  liquid  descending  slowly  through  a  liquid  with 
which  it  does  not  mix  is  spherical,  and  the  same  is  practi- 
cally true  of  a  falling  raindrop.     A  soap-bubble  consists 
of  a  thin  film  of  liquid  with  two  spherical  surfaces.     Lead 
shot,  solidifying  from  the  liquid  form  while  falling  through 
air,  are  spherical.    The  round  form  of  the  melted  end  of  a 
stick  of  sealing-wax  is  due  to  an  attempt  to  assume  the 
spherical  form.     The  hairs  of  a  camel's-hair  brush  stand 
apart  when  the  brush  is  plunged  in  water,  but  as  soon  as 
the  brush  is  drawn  out  the  film  of  water  on  it  draws  the 
hairs  together. 

158.  Surface  Tension.  —  Many  examples  of  this  tendency 
of  the  surface  to  contract  show  the  existence  of  a  contrac- 


226 


MECHANICS 


tile  force,  the  direction  of  which  is  parallel  to  the  surface. 
A  loop  of  silk  placed  on  a  film  of  a  soap  solution  is  drawn 
out  into  a  circle  as  soon  as  the 
part  of  the  film  inside  of  the  loop 
is  ruptured.  The  unbroken  part 
of  the  film  shrinks  to  a  minimum, 
and  so  causes  the  loop  of  thread 

to  enclose  the  largest  area  possible  for  a  loop  of  given 
length.  To  accomplish  this,  it  must  pull  on  the  loop  with 
a  force  parallel  to  the  surface  of  the  film. 

As  another  example,  consider  a  film  of  a  soap  solution 
on  a  rectangle  of  wire  abed,  one  side  of  which,  cd,  is  free 
to  slide  parallel  to  itself.  To  keep  cd  at  rest,  a  force  F 
away  from  ab  must  be  applied  to  it.  Hence  l 

there  must  be  a  tension  in  the  film  tending 
to  draw  ab  and  cd  together.  This  tension 
exists  only  in  the  two  surfaces,  for  the  force 
F  is  found  to  be  the  same  whether  the  film 
be  a  thick  one  or  a  thin  one.  (This  state- 
ment is  not  quite  exact  in  the  case  of  the 
thinnest  films  possible,  for  in  this  case  the 
thickness  of  the  whole  film  is  less  than  the  range  of  molec- 
ular forces,  and  this  produces  complications  which  cannot 
be  considered  here.)  Thus  the  tension  in  a  liquid  film 
does  not  increase  when  the  film  is  stretched,  whereas  the 
tension  of  ah  elastic  membrane  is  increased  by  stretching. 
The  tension  or  contractile  force  across  each  unit  of 
length  on  the  surface  of  a  liquid  is  called  the  surface 
tension  T  of  the  liquid.  The  surface  tension  of  pure 
distilled  water  at  0°  C  is  75  dynes  per  centimetre,  of  ether 
19  dynes  per  centimetre. 


F 
FIG.  97. 


FLUIDS  227 

159.  Surface  Energy.  —  When  the  surface  of  a  quantity 
of  liquid  is  increased,  more  particles  are  brought  out  into 
the  surface  layer,  and,  since  the  inward  attraction  has  to 
be  overcome  in  doing  this,  work  is  necessary  to  increase 
the  surface  area  of  a  liquid.  Thus  an  increase  of  surface 
must  result  in  an  increase  of  energy  in  the  liquid,  the 
increase  of  energy  being  proportional  to  the  increase  of 
surface,  and  therefore  called  surface  energy.  Let  the 
amount  of  this  energy  per  unit  surface  be  E.  For  an 
increase  s  in  the  surface  the  increase  of  energy  will 
be  Es. 

Consider  again  the  last  example  of  §  158.  Suppose  that 
by  the  application  of  the  force  F  or  2  Tl  the  wire  cd  is 
moved  a  distance  m  away  from  ab.  The  work  necessary 
to  do  this  is  2  Tim.  Now  Im  is  the  increase  in  each 
surface,  and  2  Im  is  the  whole  increase  of  both  sur- 
faces, say  s.  Hence  the  increase  of  energy  is  Ts,  and 
this  must  be  equal  to  Es.  Hence  T=  E,  or  the  surface 
tension  is  equal  to  the  surface  energy  per  unit  area. 
Surface  tension  can  be  calculated  from  the  force  required 
to  stretch  a  film,  and  in  various  other  ways  (§  167). 
Each  such  measurement  is  also  a  measurement  of  surface 
energy. 

It  should  be  noticed  that  the  reasoning  of  the  first  paragraph  of 
this  section  applies  also  to  solids,  so  that  in  a  solid  also  there  must  be 
a  certain  amount  of  energy  located  in  the  surface  and  proportional  to 
the  surface.  A  solid  probably  also  has  a  surface  tension,  that  is,  a 
force  along  the  surface  that  has  to  be  overcome  in  increasing  the  sur- 
face, e.g.  in  stretching  a  wire,  but  it  cannot  be  measured  owing  to  the 
fact  that  the  force  required  to  stretch  the  internal  parts  of  the  solid 
is  incomparably  greater,  and  the  two  forces  cannot  in  measurement 
be  separated. 


228  MECHANICS 

160.  Angle  of  Contact. — Where  the  surface  of  a  liquid 
meets  that  of  a  solid  it  forms  with  it  a  definite  angle  called 
the  angle  of  contact.  The  angle  of  contact  of  clean  water 
and  clean  glass  (Fig.  98)  is  0°,  that  is,  the  curved  water 
surface  is  tangential  to  the  glass.  If  the  surfaces  are  not 
clean,  the  angle  may  be  large.  For  clean-  mercury  and 


FIG.  100. 

clean  glass  (Fig.  99)  the  angle  of  contact  is  about  145°, 
but  varies  considerably  with  slight  contamination. 

The  angle  of  contact  a  can  be  measured  in  various 
ways.  The  simplest  (but  not  the  most  accurate)  is  to  tilt 
the  surface  of  the  solid  until  the  surface  of  the  liquid  is 
horizontal  (Fig.  100)  right  up  to  the  solid  surface  and  then 
measure  the  angle  of  tilt.  From  this  a  can  be  readily 
deduced. 

Our  knowledge  of  the  forces  between  molecules  is  so  imperfect 
that  we  cannot  yet  give  a  full  explanation  of  .the  curvature  of  the 
surface  of  a  liquid  in  contact  with  a  solid ;  but  the  existence  of  sur- 
face energy  (§  159)  affords  some  help.  Surface  energy  is  a  form  of 
potential  energy,  and  bodies  free  to  move  are  not  in  stable  equilibrium 
unless  their  potential  energy  is  a  minimum  (§  105).  Hence  a  liquid 
in  contact  with  a  solid  will  show  a  tendency  to  spread  over  the  latter, 
as  in  Fig.  98,  if  the  energy  of  the  surface  of  the  solid  is  less  when  the 
surface  is  covered  by  the  liquid  than  when  it  is  not  covered.  When 
the  opposite  is  the  case  the  liquid  will  tend  to  recede  and  leave  the 
solid  uncovered  as  in  Fig.  99.  The  extent  to  which  the  liquid  curves 
in  either  case  is  limited  by  the  fact  that  curvature  increases  the  free 


FLUIDS 


229 


surface  of  the  liquid  and  so  produces  an  increase  of  the  total  energy 
in  that  surface. 


TtO 


161.   Pressure   on  a   Curved   Surface  of  a   Liquid.  —  An 

elastic  band  stretched  around  a  cylinder  presses  on  the 
cylinder,  and  the  cylinder  presses  back  on  the  band  with 
an  equal  force.  To  support  the  band  the  pressure  on  its 
concave  side  must  exceed  that  on  its  convex  side.  A 
curved  liquid  surface  is  also  in  a  state  of  tension,  and  for 
equilibrium  the  pressure  on 
its  concave  side  must  exceed 
that  on  its  convex  side.  The 
difference  of  pressure  on  the 
two  sides  of  a  curved  liquid 
surface  can  be  stated  in  terms 
of  the  curvature  of  the  sur- 
face and  its  surface  tension. 

For  simplicity  we  shall  con- 
sider first  the  case  of  a  surface  curved  like  the  surface  of 
a  circular  cylinder  (this  is  the  shape  of  the  surface  of  a 
liquid  between  two  parallel  plates  standing  close  together 
in  the  liquid  —  see  Fig.  102).  Let  a  plane  perpendicular 
to  the  length  of  the  cylinder  cut  the  liquid  surface  in 
ABC  and  the  axis  of  the  cylinder  in  0.  Consider  a 
short  and  very  narrow  strip  of  the  surface  in  the  form  of 
a  curved  rectangle  of  which  ABO is  one  edge  and  denote 
the  small  angle  A  00  by  0.  The  thrust  on  each  minute 
part  of  the  strip  is  along  the  radius.  Since  0  is  small 
these  thrusts  are  practically  parallel  and  their  resultant 
equals  their  sum  and  acts  along  the  bisecter  of  0. 

If  R  is  the  radius  of  the  cylinder,  the  length  of  the 


230 


MECHANICS 


strip  (in  the  direction  of  ABO)  is  R0,  and  if  its  width 
is  w,  its  area  is  ROw.  If  the  pressure  on  the  con- 
cave side  exceeds  that  on  the  convex  side  by  p  (per  unit 
area),  the  resultant  thrust  on  the  strip  is  pROw.  The 
surface  tension  ^exerts  a  force  Tw  on  each  end  of  the  strip. 
For  equilibrium  the  lines  of  these  three  forces  must 
intersect  in  a  point  D  (§  95),  and  the  sum  of  the  com- 
ponents of  the  forces  Tw  along  DO  must  be  equal  and 
opposite  to  the  resultant  thrust.  The  component  of 
each  force  Tw  along  D  0  is  Tw  cos  J  ADC,  or  Tw  sin  ^  6. 
But  since  6  is  very  small,  we  may  put  J  6  for  sin  J  0. 
Hence 


or 


T 


162.  Level  of  Liquid  between  Two  Plates.  —  Liquid  in 
contact  with  a  plate  meets  the  latter  at  a  definite  angle, 
and  is,  therefore,  curved  upward  or  downward.  The  same 
is  true  of  a  second  plate  close  to  the 
first,  and  if  the  plates  are  close 
together,  the  two  curvatures  join 
together  to  form  a  cylindrical  sur- 
face of  radius  R.  If  the  surface  is 
concave  upward,  the  pressure  on  the 
upper  side  is  atmospheric  pressure 
^  P,  and  the  pressure  at  a  point  0 
=  just  beneath  the  curved  surface  is 

T 
FIG.  102.  P  ~~B'     The  pressure  at  a  point  A 

in  the  free  surface  outside  of  the  plates  is  P,  and 
the  pressure  at  a  point  B  between  the  plates  and  at 
the  same  level  as  A  must  also  be  P.  Hence  the 


FLUIDS  231 

pressure  at  0  is  less  than  that  at  B,  and  Q  must,  there- 
fore, be  at  a  higher  level  than  B.  Let  the  difference 
of  level  of  0  and  B  be  h.  Then  the  pressure  at  B 
equals  that  at  Q  plus  gph,  p  being  the  density  of  the 
liquid.  Hence 


h==    T  t 
gpR 

A  radius  of  the  cylinder  through  a  point  in  the  line  of 
contact  makes  with  a  line  perpendicular  to  the  two  plates 
an  angle  equal  to  the  angle  of  contact  a.  Hence,  if  d  is 
the  distance  between  the  plates,  R  cos  a  =  ^  d,  and,  there- 

fore, 0  m 

,      %T  cos  a 


The  same  formula  is  obtained  if  a  case  in  which  a  liquid 
is  depressed  between  the  plates  is  considered.  For  such 
a  liquid,  a  >  90°  and  cos  a  is  negative. 

It  has  been  supposed  in  the  above  that  h  is  measured  to  the  bot- 
tom of  the  curved  surface  between  the  plates.  Some  of  the  liquid  is 
actually  at  a  higher  level,  and  it  can  be  shown  that  for  greater  accu- 
racy h  should  be  increased  by  about  ^  d  if  a  =  0.  This  correction  is, 
however,  usually  negligible. 

163.  Liquid  between  Two  Inclined  Plates.  —  If  two  ver- 
tical rectangular  plates,  standing  in  a  liquid,  touch 
along  a  vertical  line  and  are  inclined  to  one  another  at 
a  small  angle  0,  the  separation  of  the  plates  at  any  dis- 
tance x  from  the  line  of  contact  is  xO,  and  the  plates  are 
so  nearly  parallel  that  we  may  apply  the  formula  of  §  162 


232 


MECHANICS 


to  find  the  rise  of  liquid  between  them.     Denoting  the 
elevation  at  a  distance  x  by  y, 

^2  T  cos  a 
gpxB 

2  T  cos  a 


The  right-hand  side  of  this  equation  is  a  constant  for  a 
given  inclination  of  the  plates.  Hence  xy  is  constant. 
This  is  the  characteristic  of  a  rectangular  hyperbola. 
Hence  this  is  the  curve  formed  by  the  surface  of  the 
liquid. 

Exercise  XXXVII.    Surface  Tension 

Two  sheets  of  plate  glass  are  prepared  for  a  study  of  the  rise  of 
water  between  them,  as  explained  in  §  163.    One  should  be  somewhat 

larger  than  the  other,  and  a 
tank  should  be  formed  by 
cementing  strips  of  glass  to 
the  bottom  of  the  larger. 
(Strips  of  adhesive  paper 
such  as  are  used  for  the 
edges  of  lantern  slides  may 
be  used  to  bind  the  edges  to- 
gether. These  strips  should 
be  then  covered  by  a  thin 
film  of  paraffin  wax,  applied 
melted,  to  prevent  wetting 
by  the  water,  and  the  tank 
should  be  made  water-tight 
by  coating  the  inside  edges 
with  paraffin  wax.)  The 
edge  of  the  smaller  glass  plate  that  is  to  be  in  contact  with  the  other 
plate  should  be  ground  very  smooth  and  straight  (by  rubbing  it  on 
a  sheet  of  sandpaper,  tacked  to  a  board  and  wet  with  turpentine). 


FIG.  103. 


FLUIDS  233 

Fasten  a  sheet  of  cross-section  paper  to  the  vertical  board,  tak- 
ing care  that  the  lines  are  truly  vertical  ancT  horizontal.  In  front 
of  this  mount  the  larger  glass  plate,  allowing  it  to  rest  on  a  small 
platform  clamped  to  the  board.  Wash  both  plates  and  the  inside 
of  the  tank  clean  with  chromic  acid  and  distilled  water.  Then  stand 
the  smaller  plate  in  the  tank  and  level  the  supporting  platform  until 
the  line  in  which  the  plates  meet  is  vertical.  The  spring  clips  shown 
in  the  figure  are  for  the  purpose  of  keeping  an  edge  of  the  small  plate 
tight  against  the  large  plate.  A  small  strip  of  steel  of  definite 
thickness  (about  1  mm.)  is  used  to  separate  the  other  edge  of  the 
plates. 

After  filling  the  tank  nearly  full  with  distilled  water,  allow  the 
plates  to  come  closer  together  than  they  are  intended  to  remain,  and 
then  separate  them  cautiously  until  the  strip  of  steel  can  be  inserted. 
Push  the  latter  in  some  distance,  and  the  result  will  be  a  clear, 
smooth  curve,  formed  by  the  surface  of  the  water  between  the  plates. 
Find  the  abscissae  and  ordinates  of  various  points  on  the  curve,  and 
calculate  the  values  of  xy.  The  value  of  0  is  found  from  the  thick- 
ness of  the  steel  strip  and  its  distance  from  the  edge  of  contact  of  the 
plates.  The  observations  should  be  made  as  quickly  as  possible  in 
order  that  a  (which  is  0  while  the  plates  are  well  wet)  should  not 
markedly  change.  T  should  be  calculated  from  the  mean  value  of 
xy.  (Cross-section  lines  etched  on  the  front  of  the  larger  plate  are 
better  than  the  cross-section  paper,  but  are  not  indispensable.) 

DISCUSSION 

(a)  Why  does  the  liquid  rise  between  the  plates  ? 

(&)  Explain  the  force  that  urges  the  plates  together. 

(c)  Calculate  the  pressure  where  x  =  5  cm.  and  y  =  2  cm. 

(d)  Sources  of  error  in  the  value  of  T. 

(e)  Is  the  value  found  for  T  more  probably  too  high  or  too  low? 
(/)  How  high  would  the  water  rise  if  the  smaller  plate  were 

placed  with  its  lower  horizontal   edge   in   contact  with  the   other 
plate?     (This  might  be  tried  and  the  value  of  T  deduced.) 

(#)  How  could  the  quantity  of  liquid  that  rises  between  the 
plates  be  calculated? 


234  MECHANICS 

164.   Pressure  of  Curved  Liquid  Surfaces.  —  A  cylindrical 

T 

surface  of  radius  R  and  tension  T  exerts  a  pressure  —  on  the  concave 

R 

side.  Such  a  surface  is  produced  by  bending  a  plane 
surface  once,  hence  it  is  called  a  surface  of  single  curva- 
ture. If  to  a  small  part  of  a  cylindrical  surface  a  second 
curvature  be  given  by  bending  it  in  a  direction  at  right 
angles  to  the  first  direction  of  bending,  it  will  become 
part  of  a  surface  of  double  curvature,  such  as  a  sphere, 
spheroid,  etc.  (It  will  also  be  curved  in  intermediate 
directions,  but  this  is  merely  a  consequence  of  the  two 

principal  curvatures.)     If  the  radius  of  the  second  curva- 

T 

ture  be  R' .  the  tension  will  cause  a  second  pressure  — -, 
FIG.  104.  R' 

and  the  whole  pressure  will  be 


Q   rn 

In  the  case  of  a  sphere  R  =  R'.    Hence  p  =  — •     For  a  spheroid 

ellipsoid,  etc.,  R  and  R'  are  unequal  (except  at  certain  particular 
points).  There  are  surfaces,  such  as  a  saddle-back  or  spindle,  at  any 
point  on  which  the  two  curvatures  are  in  opposite  directions,  and 
R  and  R'  therefore  differ  in  sign. 

A  spherical  soap-bubble  consists  of  a  thin  sheet  of  liquid  between 
two  contractile  spherical  surfaces  of  practically  equal  radii  R  and 
under  a  tension  T.  Hence  the  pressure  inside  must  exceed  that 

A    T* 

outside  by If  a  soap-bubble   be   formed  between  two   glass 

R 
funnels  so  that  it  is  not  spherical,  the  excess  of  internal  pressure 

must  be  2  T  f  — | j  •     If  the  small  ends  of  the  funnels  be  open  to 

the  atmosphere,  there  can  be  no  excess  of   internal  pressure,  and 

therefore   — | f  =  0,  or  R  =  —  R'.      This  accounts  for  the  spindle 

shape  of  such  a  film. 

165.   Level  of  Liquid  in  a  Capillary  Tube.  — The  surface  of 
liquid  in  a  vertical  capillary  tube  (that  is,  a  tube  of  very 


FLUIDS 


235 


small  bore)  of  circular  cross-section  is  spherical.  Hence 
the  pressure  on  the  concave  side  must  exceed  that  on  the 

2  T 
convex  side  by  — ,  where  R  is  the  radius  of  the  spherical 

-TL 

surface.  Therefore  the  liquid  will  rise  above  the  ordinary 
level  if  the  concavity  is  upward,  and  it  will  be  depressed 
if  the  concavity  is  downward.  The  elevation  or  depres- 
sion is  found  exactly  as  in  §  162,  in  fact,  it  is  only  necessary 

2  T         T 

to  substitute  —  for  — .     Hence 
R          R 


gpR 

If  the  angle  of  contact  is  a  and  if  the  radius  of  the  tube 
is  r,  then  R  cos  «  =  r.     Hence 

fr_2TeOBa 

gpr 

166.  Other  Effects  of  Surface  Tension.  —  Two  bodies,  float- 
ing close  together  on  a  liquid  that  wets  both,  apparently  attract  one 
another.  The  liquid  rises  between  them,  and  while  the  pressure  in 
the  elevated  liquid  between  them  is  less  than  atmospheric  pressure 
P,  the  pressure  at  the  same  level  on  their  other  faces  is  P.  Hence 


P- 


FIG.  105. 


FIG.  106. 


FIG.  107. 


they  are  urged  together.  If  neither  body  is  wet  by  the  liquid,  the 
liquid  is  depressed  between  them.  Thus  for  a  certain  space  between 
them  the  pressure  is  P,  while  at  the  same  depth  on  their  other  faces  the 
pressure  is  due  to  liquid  below  the  free  surface,  and  therefore  exceeds 
P.  Hence  they  are  pushed  together.  If  one  is  wet  by  the  liquid  while 
the  other  is  not,  the  form  of  the  liquid  surface  is  as  shown  in  Fig.  107, 
and  it -is  readily  seen  that  each  plate  is  urged  away  from  the  other. 


236  MECHANICS 

A  piece  of  camphor  dropped  on  clean  water  begins  to  dissolve.  At 
some  points  the  solution  proceeds  faster  than  at  other  points.  At 
places  where  the  water  is  most  polluted  by  the  camphor  the  surface 
tension  is  most  weakened,  and  thus  the  camphor  is  drawn  away  by  the 
stronger  tension  on  the  opposite  side.  Hence  rapid  and  erratic  motions 
of  the  camphor  ensue.  A  very  slight  film  of  oil  in  the  water  weakens 
the  surface  tension  so  much  that  such  motions  do  not  take  place. 

167.  Methods  of  measuring  Surface  Tension.  —  The  most 
common  method  is  to  observe  the  rise  of  the  liquid  in  a 
capillary  tube  and  use   the  equation  of   §  165.      Other 
methods  that  have  been  used  depend  on  the  downward 
pull  of  the  liquid  on  a  thin  plate  partly  immersed  in  the 
liquid,  or  on  the  pull  required  to  draw  a  plate  away  from 
the  liquid,  or  on  the  size  of  drops  falling  from  a  tube,  or 
on  the  form  of  a  drop  (of  mercury)  resting  on  a  glass 
plate.     Probably  the  most  accurate  method  is  by  observa- 
tion of  the  wave  length  and  frequency  of  small  waves  or 
ripples  on  the  surface  of  the  liquid,  for  the  propagation  of 
such  waves  depends  on  surface  tension. 

168.  Diffusion ;   Osmosis.  —  When  two  liquids  that  can 
mix  are  placed  in  contact,  the  particles  of  each  begin  to 
pass  into  the  other.     This  process  is  called  diffusion.     If 
a  vial  filled  with  a  solution  of  some  salt  (e.g.  blue  solu- 
tion  of   copper   sulphate)   be  wholly  immersed   beneath 
water  in  a  beaker,  the  salt  will  slowly  diffuse  out  of  the 
vial.     The  quantity  of  salt  that  leaves  the  vial  depends 
on  the  time,  the  strength  of  the  solution,  and  the  tempera- 
ture, and  it  is  also  markedly  different  for  different  salts  or 
dissolved  substances.     The  rate  of  diffusion  of  one  salt  is 
practically  independent  of   the  presence  of   other  salts, 
provided  there  is  no  chemical  action. 


FLUIDS  237 

Substances  can  be  roughly  classified  according  to  their 
rates  of  diffusion  into  crystalloids,  such  as  mineral  salts 
and  acids,  which  diffuse  rapidly,  and  colloids,  such  as 
starch,  albumen,  and  caramel,  which  diffuse  slowly.  The 
difference  is  probably  due  to  the  fact  that  the  molecules 
of  colloids  are  larger  and,  therefore,  move  more  slowly 
than  those  of  crystalloids.  Colloids  in  water  tend  to  form 
jellies,  which  apparently  consist  of  a  more  or  less  solid 
framework,  through  which  the  liquid  is  dispersed. 
Through  such  a  jelly,  or  colloid  membrane, 
crystalloids  can  diffuse,  while  colloids  cannot. 
Wet  parchment  or  bladder  is  a  colloid,  and 
a  mixture  of  crystalloids  and  colloids  in  water 
(e.g.  the  contents  of  the  stomach  in  cases 
of  poisoning)  can  be  separated  by  placing  the 
mixture  in  a  tube  closed  below  by  parchment 
and  dipping  it  in  a  vessel  of  water.  When 
a  colloid  membrane  separates  water  and  an 
aqueous  solution,  the  pure  water  passes  more 
readily  than  the  water  of  solution.  If,  for 
example,  a  tube  closed  below  by  parchment 
be  partly  filled  with  a  sugar  solution,  and  be  dipped  in 
water  so  that  both  liquids  are  at  the  same  level,  the  level 
will  continue  for  some  time  to  rise  in  the  tube  ;  water  par- 
ticles pass  through  the  membrane  in  both  directions,  but 
more  pass  into  the  tube  than  in  the  reverse  direction. 

Certain  membranes,  called  semipermeable  membranes, 
allow  water,  but  not  dissolved  salts,  to  pass.  A  layer  of 
ferrocyanide  of  copper  deposited  chemically  in  the  pores 
of  a  porous  earthenware  plug  that  closes  the  lower  end  of 
a  tube  is  an  example.  If  the  tube  be  filled  by  a  salt  solu- 


238  MECHANICS 

tion  and  be  immersed  in  water,  the  water  will  continue  to 
enter  the  tube  and  'will  rise  until  the  pressure  of  the  col- 
umn in  the  tube  prevents  further  inflow.  The  height  in 
the  tube  depends  on  the  particular  salt,  the  strength  of 
the  solution,  and  the  temperature  ;  the  pressure  of  the 
column  {gpli)  when  it  ceases  to  rise  is  calle'd  the  osmotic 
pressure  of  the  dissolved  salt.  The  explanation  of  the 
action  is  not  yet  certain,  but  one  interesting  law  has  been 
arrived  at,  namely,  that  the  osmotic  pressure  of  a  salt  in  a 
very  weak  solution  is  equal  to  the  pressure  which  the 
particles  would  exert  if  the  water  were  supposed  absent 
and  the  particles  were  in  the  gaseous  state. 

GASES 

169.  Gases.  —  The  shear  modulus  of  a  gas,  like  that  of  a 
liquid,  is  zero.  Hence  all  the  properties  of  fluids  that 
depend  on  the  absence  of  elasticity  of  form  are  possessed 
by  gases.  Thus  the  pressure  of  a  gas  on  any  surface  is 
perpendicular  to  the  surface  (§  139),  the  pressure  at  any 
point  is  the  same  in  all  directions  (§  141),  and  an  increase 
of  pressure  at  any  point  in  a  gas  at  rest  is  accompanied 
by  an  equal  increase  at  all  points  (§  144).  Archimedes' 
principle  (§  146)  is  also  true  of  gases.  A  balloon  is  sus- 
tained by  a  force  equal  to  the  weight  of  the  air  which  it 
displaces.  When  the  weight  of  a  body  is  to  be  found  with 
great  accuracy,  allowance  must  be  made  for  the  weight 
of  the  air  displaced  by  the  body  and  also  for  the  weight 
of  the  air  displaced  by  the  weights  used.  The  pressure 
in  a  gas  also  increases  with  the  depth,  and  the  law  of 
increase  is  the  same  as  in  the  case  of  a  liquid  (§  142).  All 
gases  are  viscous,  and  the  definition  of  the  coefficient  of 


FLUIDS 


239 


viscosity  of  a  gas  is  exactly  the  same  as  that  of  a  liquid 
(§  153). 

170.  Pressure  of  the  Atmosphere.  —  An  important  case  of 
pressure  due  to  gravity  and  depth  is  the  pressure  of  the 
atmosphere.  If  a  very  long  tube  were  supposed  to  ex- 
tend from  the  surface  of  the  earth  to  the  outer  limit  of 
the  atmosphere  (200  miles  or  more),  the  pressure  at  the 
bottom  of  the  tube  would  equal  the  weight  of  the  air  in 
the  tube.  We  cannot  calculate  the  pressure 
directly  by  means  of  the  formula  p  =  gph,  since 
the  density  is  different  at  different  heights. 
We  can,  however,  find  the  pressure  directly 
by  balancing  it  against  the  pressure  produced 
by  a  column  of  some  dense  fluid  such  as 
mercury. 

A  pressure-gauge  for  measuring  the  press- 
ure of  the  atmosphere  is  called  a  barometer. 
One  form  (Bunsen's)  consists  of  a  U-tube 
having  a  long  closed  arm  occupied  only  by 
mercury  and  a  shorter  arm  partly  occupied 
by  mercury  and  open  to  the  atmosphere.  If 
the  long  arm  is  of  sufficient  length,  there  will  be  a  vacuum 
above  the  mercury,  and  the  pressure  at  the  level  of  the 
surface  will  consequently  be  zero.  If  the  difference  of 
the  level  of  the  surfaces  in  the  two  arms  is  h  and  the 
density  of  mercury  is  p,  the  pressure  in  the  mercury  sur- 
face in  the  short  arm  is  gph,  and  this  for  equilibrium  must 
also  be  the  pressure  of  the  atmosphere. 

Another  form  of  barometer,  called  the  cistern  barometer, 
consists  of  a  straight  tube  filled  with  mercury.     The  press- 


FIG.  109. 


240  MECHANICS 

ure  at  the  surface  of  the  pool  is  atmospheric  pressure  and 
the  equal  pressure  at  the  same  level  in  the  tube  is  that 
due  to  a  column  of  mercury  equal  to  the  difference  of 
level  of  the  two  mercury  surfaces,  or  gph.  When  the 
atmospheric  pressure  increases,  the  level  of  the  mercury 
in  the  tube  rises  and  that  in  the  cistern  falls.  By  reading 
on  a  scale  etched  on  the  glass  tube  or  on  a  separate  scale 
placed  beside  the  tube,  h  may  be  found,  but  this  will  re- 
quire a  reading  of  each  of  the  two  mercury  surfaces.  In 
Fortin's  barometer  this  double  reading  is  avoided  by  using 
a  cistern  with  a  flexible  leather  bottom ;  by  adjusting  a 
screw  that  presses  on  the  leather  the  level  of  the  mercury 
in  the  cistern  may  be  brought  to  the  zero  of  the  scale. 

The  aneroid  barometer  is  a  shallow,  cylindrical,  metal 
box  exhausted  of  air ;  the  top  rises  and  falls  with  changes 
of  atmospheric  pressure,  and  its  motion  is  communicated, 
by  a  magnifying  system  of  levers,  to  an  index  that  indi- 
cates the  pressure  on  a  scale  which  is  graduated  by  com- 
parison with  a  mercury  barometer. 

171.  Corrections  of  Barometer  Reading.  —  For  many  purposes 
it  is.  necessary  to  compare  the  atmospheric  pressure  at  different  times 
and  at  different  places.  To  do  this  it  is  not  sufficient  merely  to  com- 
pare the  heights  of  the  barometer,  for,  since  P  =  gpH,  variations  in 
the  values  of  p  and  g  should  be  allowed  for.  Moreover,  H  is  measured 
on  a  scale,  the  length  of  each  unit  of  which  depends  on  the  tempera- 
ture. To  allow  for  these  differences  it  is  customary  to  calculate  from 
H  what  the  height,  say  H0,  of  the  barometer  would  have  been,  if  (the 
actual  pressure  remaining  the  same)  the  temperature  of  the  mercury 
and  the  scale  had  been  that  of  melting  ice,  or  zero  on  the  centigrade 
scale,  and  g  had  been  equal  to  the  acceleration  of  gravity  at  sea-level 
in  a  latitude  of  45°,  say  gQ.  Let  p  be  the  density  of  mercury  at  the 
actual  temperature  t°  C.  and  pQ  its  density  at  0°  C. ;  also  let  each  unit 


FLUIDS  241 

of  the  scale  equal  n  true  units  of  length  (centimetres  or  inches)  at 
t°  C.,  and  n0  true  units  at  0°  C.  Since  the  pressure  may  be  expressed 
either  as  gpHn  or  as  ^r0p0//0n0, 

~  9o    Po    no 

Now  if  X  is  the  latitude  of  the  place  of  observation  and  I  its  height 
above  sea-level  in  metres  (§  56), 

•^  =  1  -  .0026  cos  2  X  -  .0000003  I. 
<7o 

It  is  shown  in  works  on  Heat  that  (assuming  the  scale  to  be  of  brass) 

•£  =  (1  -  .000181  0  and  -  =  (1  +  .000019  *). ' 
Po  no 

Multiplying  these  factors  together  and  neglecting  the  products  of 
small  quantities,  we  get 

#0  =  (1  -  .000162  0  (1  -  .0026  cos  2  X  -  .000000  3  V)H. 

This  value  of  H0  is  in  scale  units  at  0°  C.  If  the  scale  is  correct 
at  0°  C.,  no  further  correction  is  required ;  if  it  is  not,  HQ  must  be 
multiplied  by  the  ratio  of  the  scale  unit  at  0°  C.  to  the  centimetre 
or  inch. 

A  unit  sometimes  employed  in  stating  pressures  is  the  standard 
atmosphere,  that  is,  the  pressure  of  a  column  of  mercury  76  cm.  high, 
at  0°  C.  at  sea-level  in  the  latitude  of  45°. 

Another  source  of  error  in  estimating  the  pressure  by  a  cistern 
barometer  is  curvature  of  the  surface  of  mercury  in  the  tube.  The 
curvature  being  downward,  the  surface  tension  causes  a  downward 
pressure  on  the  mercury  column,  thus  causing  it  to  be  somewhat 
depressed.  To  get  the  true  barometric  height,  the  observed  height 
must  be  corrected  by  adding  the  amount  of  this  depression.  The 
magnitude  of  this  correction  is  negligible  for  tubes  of  2.5  cm.  or  more 
in  diameter.  A  table  giving  the  amount  of  the  depression  for  tubes 
of  various  sizes  has  been  drawn  up  from  comparisons  of  various 
barometers  with  a  barometer  the  tube  of  which  is  so  large  that  the 
depression  is  negligible.  (See  the  Smithsonian  Tables,  p.  124.) 

R 


242 


MECHANICS 


FIG.  110. 


172.  Pumps.  —  In  the  common  lift  pump  water  is  raised 
by  atmospheric  pressure.  A  piston  P  moves  in  a  cylin- 
der (7,  which  is  connected  by  a  pipe  Q  to  the  water  in  the 
well.  A  valve  Vl  at  the  bottom  of  the  cylinder  and  a 
valve  FJj  in  the  piston  open  upward.  When 
P  is  raised,  Fj  opens  and  F^  closes.  Air 
from  Q  passes  into  (7,  and  the  pressure  in  Q 
being  diminished,  the  water  rises  in  Q. 
When  the  piston  is  forced  dow#,  Vl  closes 
and  air  is  forced  out  through  F^.  After  a 
few  strokes,  water  rises  into  (7,  and  when  P 
descends  the  water  passes  into  the  part  of 
the  cylinder  above  P.  Thereafter  at  each 
stroke  water  flows  out  through  the  spout  R. 
If  the  length  of  the  tube  Q  be  too  great, 
water  will  not  rise  in  the  cylinder.  Since 
mercury  will  rise  to  about  76  cm.  in  a  vacuum,  and  mer- 
cury is  13.6  times  as  dense  as  water,  water  will  rise  in  a 
vacuum  to  a  height  of  about  76  x  13.6  cm.  or  1034  cm., 
or  33.9  ft.  As  a  matter  of  fact,  the  suction 
pump  fails  at  a  height  less  than  this  ;  for, 
even  if  there  is  no  leakage  between  the 
piston  and  the  cylinder,  the  water  itself 
contains  some  air  in  solution,  and  the  air 
separating  out  causes  a  pressure  above  the 
column  of  water. 

By  means  of  the  force  pump  water  may 
be  raised  to  a  great  height.  In  this  pump 
there  is  no  valve  in  the  piston,  and  water 
is  forced  up  a  side  tube  R  as  the  piston 
descends.  A  valve  in  R  prevents  the  FIG.  in. 


FLUIDS  243 

return  of  the  water  to  O  as  the  piston  rises.  The  outflow 
through  R  takes  place  only  during  the  downward  motion  of 
P,  but  if  an  "  air  chamber  "  A  be  inserted  in  R,  the  air,  being 
constantly  under  pressure,  will  cause  a  continuous  outflow. 

173.  The  Siphon.  —  If  the  ends  of  a  U-shaped  tube  full 
of  liquid  be  closed  and  the  tube  be  then  inverted,  and  one 
end  be  immersed  in  liquid,  the  liquid  will  flow 
out  when  the  ends  are  opened,  provided  the 
end  in  air  be  at  a  lower  level  than  the  surface 


of  the  liquid  in  the  vessel.  Let  the  depth  of 
the  open  end  A  below  the  surface  of  the  liquid 
in  the  vessel  be  h.  Before  A  was  opened  the  A 

pressure  in  the  liquid  at  A  was  greater  than 
atmospheric  pressure  by  gph,  and  when  A  was  opened  the 
opposing  pressure  was  only  atmospheric  pressure.  The 
siphon  can  be  used  on  a  large  scale  for  drainage,  provided 
no  part  of  the  tube  need  be  at  a  greater  distance  above  the 
level  of  the  liquid  than  the  height  to  which  water  will  rise 
in  a  vacuum. 

174.  Boyle's  Law.  —  Let  p  be  the  pressure  and  v  the 
volume  of  a  mass  of  gas  the  temperature  of  which  is  kept 
constant.  The  product  pv  =  a  constant,  no  matter  how  much 
p  and  v  may  separately  change.  This  law,  discovered  by 
Robert  Boyle  in  1662  and  verified  by  him  both  for  pressures 
greater  than  atmospheric  pressure  and  for  pressures  less,  is 
called  Boyle's  Law.  (The  apparatus  used  by  Boyle  was  not 
essentially  different  from,  that  used  in  the  next  exercise.) 

It  is  evident  that  Boyle's  Law  may  also  be  stated  as 
follows  :  the  pressure  of  a  gas  at  constant  temperature 
varies  inversely  as  its  volume.  For  a  greater  mass  at  the 


244 


MECHANICS 


same  pressure  the  volume  will  be  proportionally  greater. 
Hence  pv  =  ^ 

m  being  the  mass  and  k  a  constant.     Now  m  -s-  v  is  the 
density  />  of  the  gas.     Hence,  for  a  constant  mass  of  gas, 

p=~kp. 

Exercise  XXXVIII.    Boyle's  Law 

Two  glass  tubes  A  and  B  (Fig.  113)  are 
mounted  on  blocks  that  can  be  clamped  at 
various  positions  along  a  vertical  scale.  Between 
each  tube  and  block  is  a  strip  of  mirror  glass. 
A  rubber  tube  connects  the  lower  ends  of  the 
glass  tubes.  In  the  upper  end  of  A  is  a  per- 
forated rubber  stopper  which  is  shellacked 
before  being  inserted.  The  perforation  in  the 
stopper  can  be  closed  by  a  round  nail  coated 
with  a  mixture  of  beeswax  and  vaseline  to 
make  it  air-tight.  The  frame  of  the  apparatus 
can  be  levelled  until  the  scale  is  vertical  as 
indicated  by  a  plumb-line. 

The  glass  tubes  are  first  brought  to  about 
the  same  level  and  mercury  is  poured  in  until 
it  about  half  fills  each  tube.  A  drying  tube 
(containing  calcium  chloride)  is  then  inserted 
into  the  perforation  in  A,  and  A  is  filled  with 
dry  air  by  alternately  raising  and  lowering  B 
several  times.  The  nail  is  then  inserted  in  A 
so  that  the  lower  end  just  appears  below  the 
cork. 

The  pressure  in  A  is  less  or  greater  than 
atmospheric  pressure  according  as  A  is  higher 
or  lower  than  B.  The  level  of  the  mercury  in 
A  and  in  B  is  read  on  the  scale  by  means  of 
a  small  T-square,  one  arm  of  which  is  pressed 


FIG.  113. 


FLUIDS  245 

against  the  framework  at  such  a  level  that  an  edge  of  the  other 
(horizontal)  arm,  its  reflection  in  the  mirror,  and  the  surface  of 
the  mercury  seem  to  coincide;  the  position  of  the  horizontal  arm 
on  the  scale  (which  it  touches)  should  be  read  with  the  greatest 
care.  The  level  of  the  lower  end  of  the  stopper  in  A  is  found  in 
the  same  way.  From  these  readings  the  pressure  in  A  and  the 
length  of  the  column  of  air  in  A  (which  is  proportional  to  the  volume 
of  the  air)  are  deduced.  These  readings  should  be  made  with  A  at 
the  top  of  the  scale  and  B  at  the  bottom,  and  then  with  A  and  B 
in  various  intermediate  positions,  and  finally  with  A  at  the  bottom  of 
the  scale  and  B  at  the  top.  From  the  readings  thus  made  the  con- 
stancy of  pv  can  be  tested. 

DISCUSSION 

(a)  Sources  of  error. 

(6)  Do  the  tubes  need  to  be  of  the  same  diameter  ? 

(c)  Why  should  A  not  be  of  very  small  bore  ? 

(rf)  Does  the  stretching  of  the  rubber  tube  affect  the  result? 

(e)  Why  should  the  air  be  dry?  Does  it  need  to  be  perfectly  dry? 
What  would  be  the  effect  of  a  film  of  water  on  the  mercury  in  A  ? 

(/)  How  could  the  volume  (and  density)  of  a  quantity  of  a 
powder  (gunpowder,  sugar,  salt)  be  found  by  placing  it  in  a  vessel 
suspended  in  A  ? 

175.  Deviations  from  Boyle's  Law.  —  For  many  gases,  such 
as  oxygen,  hydrogen,  and  nitrogen,  Boyle's  Law  is  so  nearly 
exact  that  for  most  purposes  it  may  be 
taken  as  perfectly  accurate.  Careful 
study  has,  however,  shown  that  it  is 
in  no  case  perfectly  exact.  The 
general  nature  of  the  deviations  is 
shown  by  Fig.  114.  The  continuous 
line  shows  the  connection  between 


pressure    and    volume    as    found    by 

careful  experiment,  while  the  dotted  line  indicates  what 


246  MECHANICS 

the  connection  would  be  if  Boyle's  Law  were  accurately 
followed.  These  two  curves  intersect  in  two  points  A 
and  B.  Consider  the  case  of  air,  and  suppose  that  when 
the  mass  of  air  is  in  the  condition  represented  by  the 
point  A,  its  pressure  is  one  atmosphere.  When  it  is 
brought  to  the  condition  represented  by  _B,  the  product 
of  its  pressure  and  volume  is  the  same  as  if  it  had  per- 
fectly followed  Boyle's  Law,  its  pressure  being  152  atmos- 
pheres and  its  volume  y^  °^  its  volume  at  one  atmosphere. 
The  smallest  value  of  pv  between  these  limits  occurs  at 
about  78  atmospheres  and  is  .98  of  the  product  pv  at  A 
and  B.  Other  gases  show  similar  deviations,  but  the 
details  are  different  for  different  gases. 

Very  careful  experiments  have  shown  that  the  relation 
between  the  pressure  and  volume  of  a  gas  is  more  accu- 
rately represented  by  the  formula  (due  to  van  der  Waals), 


—  ](v  —  5)  =  a  constant, 


a  and  b  being  small  numbers  the  magnitudes  of  which  are 
different  for  different  gases. 

176.  Modulus  of  Elasticity  of  a  Gas.  —  While  the  shear 
modulus  of  a  gas  is  zero  (§  138),  the  bulk  modulus  has 
a  definite  value,  and  it  is  accordingly  the  latter  that  is 
always  meant  when  the  modulus  of  elasticity  of  a  gas  is 
referred  to.  Let  the  pressure  and  volume  of  a  mass  of 
gas  be  p  and  v,  and  suppose  that  a  small  increase  of  press- 
ure a  produces  a  decrease  ft  in  the  volume.  Then,  by 
Boyle's  Law, 


FLUIDS  247 

Neglecting  the  product  a/3  of  the  two  small  quantities  « 
and  ft,  the  above  reduces  to 

va  —  p/3  =  0. 

Now  the  bulk  modulus  is  the  ratio  of  the  increase  of 
pressure  to  the  fractional  decrease  of  volume,  that  is,  the 
ratio  of  VL  to  — ,  and  from  the  above  this  is  equal  to  p. 

Hence  the  modulus  of  elasticity  is  numerically  equal  to 
the  pressure.  It  should  be  noticed  that  the  temperature 
is  supposed  to  be  constant. 

177.  Kinetic  Theory  of  Gases.  —  The  properties  of  gases 
are  consistent  with  the  view  that  a  gas  consists  of  particles 
moving  with  great  velocities  in  the  space  occupied  by  the 
gas,  that  the  impacts  of  these  particles  on  the  walls  of  the 
containing  vessel  produce  the  pressure  of  the  gas,  and 
that  the  coefficient  of  restitution  at  each  impact  is  1 
(§  112).  The  evidence  for  this  theory  is  the  fact  that  it 
will  explain  the  chief  properties  of  gases,  and  as  an  illus- 
tration we  shall  show  that  it  leads  to  Boyle's  Law. 

Let  a  rectangular  vessel  of  edges  #,  5,  and  c  contain  a 
single  gas,  and  let  the  mass  of  each  particle  be  m,  and  let 
the  number  of  particles  in  unit  volume  be 
w,  so  that  if  N  is  the  whole  number  of 
particles  in  the  vessel,  N=  nabc.  For 
brevity  denote  the  two  faces  perpendicular 
to  a  by  A  and  A'.  Resolve  the  velocity  V 
of  any  particle  into  components  w,  v,  and  w  FlG-  115- 
in  the  directions  of  a,  5,  and  <?,  respectively.  The 
pressure  on  A  depends  only  on  the  ^-components  of 
the  velocities  of  the  particles.  After  striking  A  with 


248  MECHANICS 

a  velocity  w,  a  particle  rebounds  with  a  velocity  —  u, 
and  its  change  of  velocity  is  2  M.  Hence  its  change  of 
momentum  is  2  mu,  and  this  is  therefore  the  momentum 
received  by  A.  The  particle  traverses  the  distance 

a  between  A  and  A'  in  time  -,  and  after  rebounding  from 

U  2a 

A'  it  again  strikes  A  at  a  time  -  -  after  the  first  impact. 

Hence  it  impinges  on  A  ^-  times  per  second,  and  the 
total  momentum  it  imparts  to  A  in  a  second  is,  therefore, 

n 

mu  .  For  any  other  particle  u  is  different,  but  m  and  a 
are  the  same.  Hence  the  total  momentum  imparted  to  A 
in  a  second,  that  is,  the  whole  force  on  A,  is  —  Zu2.  Divid- 
ing this  by  the  area  be  of  A  we  get  for  the  pressure  p 


on  A  the  expression  ^-^u2.     Now  N=n-abc.      Hence 


-  — 

p  =  mn--  =  mnu2,  where  u2  is  the  mean  value  of  u2  for 

all  the  particles.  But  V2  —  u2  +  v2  +  w2,  and,  since  the 
number  of  particles  is  very  large  and  they  are  moving  at 
random,  the  mean  values  of  w2,  v2,  and  w2  are  equal. 
Hence,  denoting  the  mean  value  of  V2  by  V2,  V2  =  3  v2. 

Hence  p  =  ^  mn  V2 


since  mn  is  the  whole  mass  in  unit  volume,  that  is,  the 
density  p  of  the  gas.  Now  there  is  reason  to  believe 
that  V2  is  constant,  provided  the  temperature  does  not 
change.  Hence  we  see  that  at  constant  temperature  the 
pressure  of  a  gas  is  proportional  to  its  density,  which  is 
Boyle's  Law. 


FLUIDS 


249 


Since  p  and  p  can  be  measured  experimentally,  F2  can  be 
deduced.  From  this  the  mean  velocity  (or  more  accurately 
the  square  root  of  the  mean  squared-velocity)  can  be 
deduced.  For  hydrogen  at  0°  C.  it  is  1843  metres  per 
second,  and  for  carbonic  acid  392  metres  per  second. 

If  several  different  gases  be  present  in  the  same  en- 
closure, each  will  exercise  its  own  separate  pressure,  and 
the  total  pressure  p  will  be  the  sum  of  the  separate  or 
partial  pressures  pv  p^  •••  of  the  different  gases.  The 
statement,  which  like  Boyle's  Law  is  not  perfectly  exact, 
is  called  Dalton's  Law. 


178.  Air-pumps.  —  For  removing  gas  from  a  vessel, 
pumps  are  used,  which  are  identical  in  principle  with  the 
suction  pump  used  for  water  (Fig.  110).  The  efficiency 
of  such  pumps  is  limited  by  a  variety  of 
defects.  Some  gas  leaks  in  between  the 
piston  and  the  cylinder,  and  the  piston 
cannot  be  brought  to  such  close  contact 
with  the  bottom  of  the  cylinder  as  to  expel 
all  the  gas  between  them.  For  these  and 
other  reasons,  when  a  very  high  degree  of 
exhaustion  is  required,  pumps  on  a  different 
principle  are  used. 

In  the   Geissler-Toepler  pump,  mercury 
in  a  glass  tube  is  used  instead  of  the  piston   H 
and    cylinder    of    the    mechanical    pump. 
Fig.  116  shows  its  general  principle.     The 
tube  C  is  connected  with  the  vessel  to  be 
exhausted.     The   long   flexible   rubber   tube   G-  connects 
the  mercury  reservoir  H  to   the   glass   bulb  A.     When 


FIG.  lie. 


250  MECHANICS 

If  is  raised,  mercury  enters  A,  B,  and  (7,  and  seals 
the  connection  between  A  and  0.  As  If  is  further 
raised,  the  gas  in  A  is  driven  out  through  D,  and  escapes 
from  beneath  the  mercury  in  the  vessel  E.  When  H 
is  lowered,  the  mercury  rises  in  E  and  prevents  the 
entrance  of  air,  and  gas  is  drawn  in  through  0  and  fills 
the  bulb  A.  When  H  is  raised  again,  the  gas  in  'A  is 
expelled  through  Z>,  and  so  the  process  is  continued. 
When  nearly  all  the  gas  in  the  vessel  connected  with  0 
has  been  removed,  there  will  be  very  little  pressure  in  A 
and  the  tube  connected  with  it.  Hence  the  mercury  will 
rise  to  nearly  barometric  height  in  Z>,  which  must  be 
76  cm.  or  more  in  length.  To  completely  expel  the  gas 
from  A,  B,  and  D,  H  must  be  raised  to  such  a  height  that 
the  mercury  will  pass  over  into  D ;  and  to  prevent  mercury 
passing  out  through  (7,  the  latter  must  be  very  long.  The 
purpose  of  B  is  to  prevent  danger  of  A  being  broken  by 
the  sudden  inrush  of  gas  through  0  as  H  is  lowered. 

Mercury  pumps  are  used  to  exhaust  incandescent  lamp 
bulbs.  By  such  pumps  it  is  possible  to  reduce  the  press- 
ure in  a  vessel  to  .00001  mm.  of  mercury.  For  measuring 
such  low  pressure  a  special  gauge  (the  Macleod  gauge) 
is  used. 

179.  Effusion  of  Gases.  —  The  motion  of  a  gas  escaping 
through  a  tube  is  opposed  by  the  friction  or  viscosity  of 
the  gas,  and  the  same  is  true  when  the  escape  is  through 
an  aperture  so  narrow  compared  with  its  length  that  it 
may  be  regarded  as  a  tube.  But  when  the  escape  is 
through  an  aperture  in  a  thin  wall,  the  effect  of  viscosity 
is  very  small  and  may  for  many  purposes  be  neglected. 


FLUIDS  251 

In  this  case  the  kinetic  energy  gained  by  the  escaping  gas 
is  equal  to  the  work  done  by  the  pressure  in  the  vessel  in 
causing  the  outflow.  It  would  not  affect  the  motion  and 
it  will  simplify  the  problem  if  we  suppose  that  a  friction- 
less  tube  of  the  same  cross-section  as  the  aperture  is  con- 
nected to  the  aperture,  and  that  the 
escaping  gas  drives  a  weightless 
piston  along  the  tube.  If  the  piston 
moves  from  B  to  C  in  a  second, 
BO  equals  the  velocity  v  of  the 

escaping  gas,  and  if  s  is  the  cross-section  of  the  tube 
and  aperture,  the  mass  of  gas  that  escapes  in  a  second  is 
B  0  •  s  -  p  or  vsp,  and  its  kinetic  energy  is  J  •  vsp  •  v2.  The 
work  done  by  the  pressure  p  that  causes  the  outflow 
(that  is,  the  excess  of  the  pressure  in  the  vessel  over  the 
external  pressure)  is  psBO  or  psv.  Equating  the  work 
done  and  the  kinetic  energy  gained,  we  get 


p 

Hence  under  equal  pressures  the  velocities  of  escape  of 
different  gases  are  inversely  as  the  square  roots  of  their 
densities.  This  is  the  basis  of  Bunsen's  method  of  com- 
paring the  densities  of  gases. 

If  the  pressure  p  were  supposed  due  to  the  weight  of  a 
column  of  the  gas  of  uniform  density  p  and  height  A,  we 
would  have  p  =  gpli.  If  this  be  substituted  in  the  above 
formula,  it  will  be  identical  with  the  formula  for  the  out- 
flow of  a  liquid  (§  150). 

180.  Diffusion  of  Gases.  —  If  two  equal  bottles  contain- 
ing different  gases  be  placed  mouth  to  mouth,  each  gas 


252  MECHANICS 

will  pass  into  the  other  at  a  very  rapid  rate,  and  after  a 
short  time  each  gas  will  be  equally  divided  between  the  two 
bottles.  The  result  is  independent  of  gravity  and  is  the 
same  whether  the  bottle  containing  the  heavier  gas  (e.g. 
carbonic  acid)  is  above  or  below  that  containing  the  lighter 
gas  (e.g.  hydrogen).  This  process  of  diffusion  accounts 
for  the  fact  that  the  proportions  of  oxygen  and  nitrogen 
in  the  atmosphere  are  practically  the  same  everywhere. 

Diffusion  also  takes  place  when  two  gases  are  separated  by  a  porous 
partition  such  as  unglazed  earthenware.  Lighter  gases  pass  more 
rapidly  than  heavier  gases  through  such  a  partition,  but  the 
final  result  is  the  same  as  if  the  partition  were  absent.  If 
one  end  of  a  glass  tube  be  sealed  into  a  small  dry  earthen- 
ware jar  (such  as  is  used  in  a  Bunsen's  battery)  while  the 
other  end  is  immersed  in  water  in  a  beaker,  and  if  the  jar 
be  covered  by  an  inverted  beaker  into  which  coal  gas  is 
allowed  to  stream,  air  will  be  forced  out  through  the  lower 
end  of  the  tube,  owing  to  the  lighter  gas  entering  through 
the  porous  jar  faster  than  the  air  escapes  through  it.  When 
the  jar  has  thus  become  full  of  a  mixture  of  air  and  gas,  if 
the  large  beaker  be  removed  so  that  the  porous  jar  is  now 
surrounded  by  air,  water  will  rise  in  the  tube,  owing  to  the 
gas  within  the  jar  escaping  more  rapidly  than  the  air  enters. 
The  difference  in  the  rates  of  diffusion  of  different  gases 
through  a  porous  partition  is  the  basis  of  a  method  of  separating 
mixed  gases. 

REFERENCES 

Poynting  and  Thomson's  "  Properties  of  Matter." 
Gray's  «  Treatise  on  Physics,"  Vol.  I. 
Tait's  "  Properties  of  Matter." 

"  The  Laws  of  Gases  "  (the  original  papers  of  Boyle  and  Amagat), 
in  "  Scientific  Memoirs  "  series. 

Risteen's  "  Molecules  and  the  Molecular  Theory  of  Matter." 
Boys's  "  Soap  Bubbles." 


PROBLEMS 


(The  student  is  recommended  to  use  C.  G.  S.  absolute  units  when  the 
problem  is  stated  in  metric  units  and  F.  P.  S.  gravitational  units  when  the 
problem  is  stated  in  British  units.) 

1.  Find  the  magnitude  and  direction  of  the  resultant  of  two  displace- 
ments of  magnitude  12  and  15  in  directions  that  differ  by  30°. 

2.  Find  by  the  analytical  method  the  magnitude  and  direction  of  the 
resultant  of  three  velocities,  10  east,  20  north,  and  16  southwest. 

3.  A  body  starts  with  a  velocity  of  20  ft.  per  second  and  has  an 
acceleration  of  32  ft.  per  second2  in  the  direction  of  motion.      What  is 
its  velocity  and  distance  from  the  starting-point  1,  3,  and  6  sec.  after 
starting  ? 

4.  A  body  starts  with  a  velocity  of  50  cm.  per  second,  and  in  6|  sec. 
has  acquired  a  velocity  of  102  cm.  per  second.    What  is  its  acceleration 
and  how  far  has  it  travelled  ? 

5.  A  steamship  is  moving  due  east  with  a  velocity  of  20  mi.  an 
hour,  and  to  the  passengers  the  wind  seems  to  blow  from  the  north  with 
a  velocity  of  12  mi.  an  hour.    Find  the  actual  velocity  and  direction  of 
the  wind. 

6.  A  train  having  a  speed  of  70  km.  per  hour  is  brought  to  rest  in  a 
distance  of  600  m.     What  is  its  acceleration  ? 

7.  A  body  slides  down  a  smooth  inclined  plane  and  in  the  third  sec- 
ond travels  110  cm.     What  is  the  inclination  of  the  plane  ? 

8.  A  carriage  wheel,  1  m.  in  diameter,  makes  200  revolutions  per 
minute.    What  is  the  instantaneous  speed  of  a  point  on  the  tire  (1)  when 
it  is  1  m.  from  the  ground  ;  (2)  when  it  is  0.5  m.  from  the  ground  ; 
(3)  when  it  is  on  the  ground? 

9.  A  body  is  projected  at  an  angle  of  60°  with  the  horizontal  with  a 
velocity  of  40  m.  per  sec.    How  long  will  it  move  and  how  high  will 
it  rise  ?     When  and  where  will  it  again  meet  the  horizontal  plane  through 
the  starting-point  ? 


254  PROBLEMS 

10.  A  railway  train  rounds  a  curve  of  1000  ft.  radius,  with  a  speed 
of  50  miles  per  hour.     What  is  its  acceleration  ? 

11.  How  much  is  the  acceleration  of  a  falling  body  at  the  equator  de- 
creased by  the  rotation  of  the  earth  (assume  the  radius  of  the  earth  to 
be  4000  mi.)? 

12.  A  flywheel,  making  10  revolutions  per  second,  comes  to  rest  in  1 
min.    Find  its  angular  acceleration  and  the  number  of  revolutions. 

13.  Express  980  cm.  per  second2  in  kilometres  per  minute.2 

14.  A  force  of  1000  dynes  acts  on  a  mass  of  10  g.  for  1  min.     Find 
the  velocity  acquired  and  the  distance  traversed. 

15.  In  what  time  will  a  force  of  5  kg.  weight  moves  a  mass  of  10  kg.  a 
distance  of  50  m.  ?    What  will  be  the  velocity  at  the  end  of  10  sec.  ? 

16.  What  force  must  act  on  a  mass  of  50  kg.  to  increase  its  velocity 
from  100  cm.  per  second  to  200  cm.  per  second  while  the  body  passes  over 
50m.? 

17.  Find  the  resistance  when  a  body  weighing  20  oz.,  projected  along 
a  rough  table  with  a  velocity  of  48  ft.  per  second,  is  brought  to  rest  in 
5  sec. 

18.  What  constant  force  will  lift  a  mass  of  50  Ib.  200  ft.  vertically  in 
10  sec.  ?    Find  the  velocity  at  the  end  of  that  time. 

19.  What  pressure  will  a  man  who  weighs  70  kg.  exert  upon  an  ele- 
vator descending  with  an  acceleration  of  100  cm.  per  second 2  ?    If  ascend- 
ing with  the  same  acceleration  ? 

20.  One  minute  after  leaving  a  station  a  train  has  a  velocity  of  30 
mi.  an  hour,  what  is  the  ratio  of  the  resultant  horizontal  force  to  the 
weight  of  the  train  ? 

21.  A  train  is  moving  at  a  rate  of  20  mi.  an  hour  when  the  steam  is 
shut  off.     If  the  resistance  of  friction  amounts  to  -fo  of  the  weight  of  the 
train,  how  far  will  it  run  up  a  5°  incline  ? 

22.  A  body  weighing  2  kg.  rests  on  a  table  and  is  acted  on  by  a  force 
of  8  kg.  weight,  making  an  angle  of  40°  with  the  horizontal.    What  is  the 
total  pressure  on  the  table  ? 

23.  The  diameter  of  the  bore  of  a  gun  is  10  in.  and  the  explosion  of 
the  powder  exerts  a  pressure  of  30,000  Ib.  weight  per  square  inch  on  the 
end  of  a  projectile  which  weighs  372  Ib.      If  the  pressure  of  the  powder  is 
constant,  and  the  projectile  moves  to  the  muzzle  in  y^  of  a  second,  what 
is  the  velocity  of  the  projectile  ? 


PROBLEMS  255 

24.  An  iron  ball  of  40  kg.  mass  falls  100  cm.  vertically  and  drives  a 
nail  2  cm.  into  a  plank.     What  is  the  pressure  on  the  nail  if  it  be  sup- 
posed constant  ? 

25.  A  mass  of  5  kg.  rests  on  an  inclined  plane  which  has  a  length  of 
30  cm.  and  a  height  of  2  cm.     Find  the  pressure  on  the  plane  and  the 
resistance  of  friction. 

26.  The  ends  of  a  cord  15  ft.  long  are  attached  to  two  pegs  at  the  same 
level  and  10  ft.  apart.    If  a  mass  of  100  Ib.  is  attached  to  the  middle  of 
the  cord,  what  is  the  force  on  each  peg  ? 

27.  A  weight  of  100  Ib.  hangs  at  the  end  of  a  cord.     What  horizontal 
force  will  deflect  the  cord  30°,  and  what  will  be  the  tension  in  the  cord  ? 

28.  A  rapid-firing  gun  delivers  in  a  second  10  projectiles  of  1  Ib.  each 
with  a  speed  of  2000  ft.  per  second.     What  force  is  required  to  hold  the 
gun  at  rest  ? 

29.  A  baseball  weighing  12  oz.  and  moving  with  a  velocity  of  50  ft. 
per  second  is  struck  squarely  by  a  bat  and  given  a  velocity  of  100  ft.  per 
second  in  the  opposite  direction.    If  the  contact  lasts  .005  sec.,  what  is 
the  average  force  ? 

30.  If  the  train  in  example  10  weighs  500  T.,  what  is  the  total  out- 
ward pressure  on  the  rails  ? 

31.  A  skater  describes  a  circle  of  10  m.  radius  at  a  speed  of  5  m.  per 
second.     With  what  force  do  his  skates  press  on  the  ice  ? 

32.  A  rod  of  10  kg.  mass  and  100  cm.  in  length  revolves  about  an  axis 
through  one  end,  making  10  revolutions  per  second.     Find  the  pull  on 
the  axis. 

33.  A  falling  mass  of  200  g.  is  connected  by  a  string  to  a  mass  of 
1800  g.  lying  on  a  smooth  horizontal  table.      Find  the  acceleration  and 
the  tension  of  the  string. 

34.  At  the  foot  of  a  hill,  a  toboggan  has  a  velocity  of  20  ft.  per 
second.     If  it  slides  120  ft.  on  the  horizontal,  what  is  the  coefficient  of 
friction  ? 

35.  Eeduce  a  force  of  20  Ib.  weight  to  dynes. 

36.  A  cord  passes  over  two  pulleys  and  through  a  third  movable 
pulley  between  them,  and   is  vertical  where  not  in   contact  with  the 
pulleys.      To  one  end  of  the  cord  a  mass  of  20  kg.  is  attached  and  to 
the  other  a  mass  of  10  kg.  and  the  movable  weighs  5  kg.     Find  the 
accelerations  of  the  masses. 


256  PROBLEMS 

37.  What  is  the  period  of  vibration  of  a  mass  of  1  kg.  attached  to  a 
spiral  spring  if  an  additional  mass  of  100  g.  stretches  the  spring  0.3  cm. 
farther  ? 

38.  A  man  presses  a  tool  on  a  grindstone  of  1  m.  diameter  with  a 
force  of  10  kg.  weight.     If  the  coefficient  of  friction  is  0.2,  what  force  at 
the  end  of  a  crank  arm  40  cm.  in  length  will  turn  the  stone  ? 

39.  A  disk  of  500  g.  mass  and  20  cm.  in  diameter  acquires  in  10 
sec.  a  linear  velocity  of  30  m.  per  second,  in  the  direction  of  its  axis 
and  an  angular  velocity  of  2  rotations  per  second  about  its  axis.    What 
forces  acted  on  it  ? 

40.  Find  the  centre  of  mass  of  20,  30,  24,  and  60  g.  at  the  corners  of 
a  square. 

41.  Out  of  a  circular  disk  16  cm.   in  diameter  a  circle   12  cm.  in 
diameter  and  tangential  to  the  larger  is  cut.     Where  is  the  centre  of 
mass  of  the  remainder  ? 

42.  Two  cylinders  of  the  same  material,  each  20  cm.  in  length  and  12 
and  6  cm.  in  diameter  respectively,  are  joined  so  that  their  axes  coincide. 
Find  the  centre  of  mass  of  the  whole. 

43.  An  iron  cylinder  30  cm.  in  diameter  and  of  5  kg.  mass  rolls  down 
a  plane  20  ft.  long  inclined  at  30°  to  the  horizontal.    What  linear  velocity 
does  it  acquire  ? 

44.  Find  the  resultant  of  parallel  forces  20,  40,  and  —30  applied  at  the 
corners  of  an  equilateral  triangle  of  10  cm.  side. 

45.  A  body  is  moved  from  rest  without  friction  by  a  force  that  in- 
creases uniformly  with  the  distance  traversed  from  10  Ib.  weight  to  80 
Ib.  weight.    Draw  a  diagram  of  work  done  and  find  the  kinetic  energy 
acquired,  the  total  distance  traversed  being  10  ft. 

46.  A  spiral  spring  is  attached  to  a  50  kg.  weight.    What  work  is  done 
if  the  increase  of  length  of  the  spring  is  20  cm.  when  the  weight  is  just 
lifted  ? 

47.  A  lever  20  in.  long  is  used  to  turn  a  screw  with  a  pitch  of  £  in. 
If  a  force  of  80  Ib.  weight  is  applied  to  the  lever,  what  force  will  the 
screw  exert  ? 

48.  Two  uniform  beams  each  24  ft.  long  and  of  100  Ib.  mass  are  in 
contact  at  their  upper  ends,  while  their  lower  ends  rest  on  two  vertical 
walls  of  the  same  height  and  36  ft.  apart.     Find  the  horizontal  thrust 
on  each  wall. 


PROBLEMS  257 

49.  A  runner  has  a  record  of  10  sec.  for  100  yd.,  6|  sec.  for  60  yd.,  and 
4f  sec.  for  40  yd.     What  can  be  deduced  as  to  the  horse-power  at  which 
he  works  in  running,  if  he  weighs  140  Ib.  ? 

50.  A  runner  can  run  100  yards  on  the  horizontal  in  10  sec.  and 
the  same  distance  uphill  with  a  rise  of  32  ft.   in  17.5  sec.    At  what 
horse-power  does  he  work,  if  his  weight  is  145  Ib.  ? 

51.  A  cable  100  m.  long  and  of  50  Ib.  mass  hangs  vertically  from  a 
viaduct.     How  much  work  will  be  expended  in  raising  it  ? 

52.  If  the  connection  of  the  rod  to  its  axis  in  problem  32  should  break, 
how  would  the  rod  move  ? 

53.  A  grindstone  weighs  75  kg.  and  has  a  diameter  of  1  m.     How 
much  energy  is  stored  in  it  when  it  makes  300  revolutions  per  minute  ? 

54.  When  a  hoop  rolls  on  a  rough  plane,  what  is  the  ratio  of  kinetic 
energy  of  rotation  to  that  of  translation  ? 

55.  Calculate  the  activity  of  an  engine  that  raises   1,000,000  gal. 
(each  =  10  Ib.)  of  water  in  8  hr.  from  a  depth  of  125  ft. 

56.  The  top  of  a  table  a  metre  square  projects  5  cm.  beyond  the  legs. 
If  the  table  weighs  10  kg.,  what  weight  hung  from  a  corner  will  over- 
turn it  ? 

57.  The  distance  between  the  centre  of  the  moon  and  that  of  the  earth 
is  CO  times  the  radius  of  the  earth,  and  the  mass  of  the  earth  is  82  times 
that  of  the  moon.     Find  their  centre  of  mass. 

58.  What  is  the  horse-power  of  a  locomotive  that  gives  a  train  of  200 
T.  a  velocity  of  30  mi.  an  hour  in  a  distance  of  1000  ft.  up  an  incline  of  1 
in  1000,  the  total  resistance  of  friction  being  15  Ib.  weight  per  ton  ? 

59.  What  is  the  period  of  vibration  of  a  disk  20  cm.  in  diameter  sus- 
pended on  a  horizontal  axis  perpendicular  to  the  disk  and  attached  to 
the  rim  ? 

60.  What  is  the  period  of  vibration  of  a  uniform  rod  1  m.  long  about  a 
horizontal  axis  10  cm.  from  one  end  ?    About  what  other  points  is  the 
period  of  vibration  the  same  ? 

61.  The  area  of  the  "  water-line  "  of  a  ship  is  3000  sq.  ft.    What  depth 
will  the  ship  sink  in  fresh  water  if  100  T.  be  placed  in  it  ? 

62.  How  much  will  the  above-mentioned  vessel  rise  when  it  passes  into 
salt  water  of  density  1.026  ? 

63.  The  density  of  a  body  is  2  and  m  air  of  density  .0013  it  weighs 
100.00  g.     What  is  its  true  weight,  the  density  of  the  weights  being  9  ? 


258  PROBLEMS 

64.  When  carried  from  the  ground  to  the  roof  of  a  building  a  barom- 
eter falls  1.5  mm.     What  is  the  height  of  the  building  ? 

65.  A  mass  of  copper,  suspected  of  being  hollow,  weighs  523  g.  in 
air  and  447.5  g.  in  water.     What  is  the  volume  of  the  cavity  ? 

66.  The  specific  gravity  of  ice  is  .918  and  that  of  sea  water  1.026. 
What  is  the  total  volume  of  an  iceberg  of  which  700  cu.  yd.  is  exposed  ? 

67.  A  block  of  wood  weighing  1  kg.,  the  density  of  which  is  0.7,  is  to 
be  loaded  with  lead  so  as  to  float  with  0.9  of  its  volume  immersed.    What 
weight  of  lead  is  required  (1)  if  the  lead  is  immersed  ?  (2)  if  it 'is  not  im- 
mersed ? 

68.  A  body  A  weighs  7.55  g.  in  air,  5.17  g.  in  water,  and  6.35  g.  in 
a  liquid  B.     Find  the  density  of  A  and  that  of  B. 

69.  A  retaining  wall  3  m.  wide  and  40  m.  long  is  inclined  at  40°  to  the 
horizontal.     Find  the  total  pressure  against  it  in  kilogrammes   weight 
when  the  water  rises  to  the  top. 

70.  How  far  will  water  be  projected  horizontally  from  an  aperture 
3  m.  below  the  level  of  water  in  a  tank  and  10  m.  above  the  ground  ? 

71.  The  surface  tension  of  a  soap-bubble  solution  is  27.45  dynes  per 
cm.     How  much  greater  is  the  pressure  inside  a  soap-bubble  of  3  cm. 
radius  than  in  the  outer  air  ? 

72.  If  a  submarine  boat  weighed  50  tons  and  displaced  3000  cu.  ft. 
when  immersed,  how  much  water  would  it  have  to  take  in  to  sink  ? 

73.  A  cylindrical  diving-bell  2  m.  in  height  is  lowered  until  the  top  of 
the  bell  is  6  m.  below  the  surface  of  the  water.     How  high  will  the  water 
rise  in  the  bell  if  the  height  of  the  barometer  is  76  cm.  ?     What  air  pres- 
sure in  the  bell  would  keep  the  water  out  ? 

74.  A  Fortin  barometer  reads  73  cm.  at  a  point  150  m.  above  sea- 
level  in  latitude  41°,  at  a  temperature  of  21°  C.     Reduce  the  reading  to 
0°  C.  at  sea-level  in  latitude  45°. 

75.  A  wire  300  mm.  long  and  1  mm.  in  diameter  is  stretched  1  mm.  by 
a  weight  of  3000  g.     Calculate  Young's  modulus. 

76.  To  a  wire  100  cm.  long  and  0.24  mm.  in  diameter  a  disk  whose 
moment  of  inertia  is  400  g.  cm.2  is  attached.     The  period  of  torsional  vi- 
brations is  8  sec.     Calculate  the  shear  modulus. 

77.  A  piece  of  shafting  10  m.  long  and  of  5  cm.  radius  is  twisted  1°  by 
a  certain  moment.     How  may  the  shaft  be  changed  so  that  the  twist  will 
be  30'  ? 


TABLES 


CONVERSION  TABLE 


1  cm.        =  0.3937  in. 
1  sq.  cm.  =  0.1550  sq.  in. 
1  cc.          =  0.0610  cu.  in. 
1  kg.         =  2.205  Ibs. 
1  gal.        =  4543  cc. 

ACCELERATION  0 
(In  C.  G. 
Boston     ...     -     -     980.389 

1  inch     =  2.540  cm. 
1  sq.  in.  =  6.451  sq.  cm. 
1  cu.  in.  =  16.386  cc. 
1  Ib.        =  435.6  gm. 
1  litre     =  1.7608  pints 

F  FALLING  BODY 

S.  units) 

Philadelphia    .     .     .     980.182 
San  Francisco  .     .     .     979.951 
St.  Louis      ....     979.987 
Terre  Haute     .     .     .     980.058 
Washington      .     .     .     980.100 

Chicago            . 

.     .     980.264 

Cincinnati  .     . 
Cleveland    .     . 
Denver 

.     .     979.990 

.     .     980.227 
.    979  595 

Berlin      .     .     . 

.     .     981.240 

Paris   

.     980.960 

C  Equator  •     •     • 

.     .     978.070) 

(Pole     

.     983.110) 

Greenwich  .     . 
Hammerfest     . 

Aluminium 
Brass  (about)  . 
Copper     ... 

.     .     981.170 
.     .    982.580 

DEN 

.     .        2.60 
.     .        8.50 
.     .        8  92 

Rome  

.     980.310 

.     980.852 

SITY 

Iron  (cast)  .     .     . 
Iron  (wrought) 
Lead 

7.40 
7.86 
11.30 

Gold    .... 

.     .      19.30 

Platinum      .     .     . 
Silver  . 

21.50 
10.53 

259 


260 


TABLES 


ELASTIC   CONSTANTS 
(Rough  averages;  in  C.  G.  S.  units) 


Shear 
Modulus 

Young's 
Modulus 

Bulk 
Modulus 

Parson's 
Ratio 

Copper  

4  X  1011 

11  X  1011 

17  X  1011 

.30 

Glass 

2  x  1011 

6  x  1011 

4  X  1011 

.23 

Iron  (wrought)  .... 
Lead      

7  x  1011 
.2  x  1011 

19  x  1011 
1  x  1011 

15  x  1011 
4  x  1011 

.30 
.37 

Steel                      .     . 

8  x  1011 

23  x  1011 

17  x  1011 

29 

VISCOSITY      . 
(In  C.  G.  S.  units ;  at  20°  C.) 


Alcohol 0.0011 

Ether  .  .  0.0026 


Glycerine 
Water  . 


.   8.0 
.   0.010 


SURFACE   TENSION 
(In  C.  G.  S.  units ;  at  20°  C.) 


Alcohol 21 

Ether  .  17 


Mercury 450 

Water .  74 


ANGLE   OF  CONTACT 


Alcohol 0° 

Ether  .  16° 


Mercury  (about)   .     . 
Water  (about)  .     .     . 


145° 

0° 


TABLES 


261 


TRIGONOMETRICAL  RATIOS 


Angle 

Eadians 

Sine 

Tangent 

Cotangent 

Cosine 

0° 

0 

0 

0 

oo 

1 

1.5708 

90° 

1 

.0175 

.0175 

.0175 

57.2900 

.9998 

1.5533 

89 

2 

.0349 

.0349 

.0349 

28.6363 

.9994 

1.5359 

88 

3 

.0524 

.0523 

.0524 

19.0811 

.9986 

1.5184 

87 

4 

.0698 

.0698 

.0699 

14.3006 

.9976 

1.5010 

86 

5 

.0873 

.0872 

.0875 

11.4301 

.9962 

1.4835 

85 

6 

.1047 

.1045 

.1051 

9.5144 

.9945 

1.4661 

84 

7 

.1222 

.1219 

.1228 

8.1443 

.9925 

1.4486 

83 

8 

.1396 

.1392 

.1405 

7.1154 

.9903 

1.4312 

82 

9 

.1571 

.1564 

.1584 

6.3138 

.9877 

1.4137 

81 

10 

.1745 

.1736 

.1763 

5.6713 

.9848 

1.3963 

80 

11 

.1920 

.1908 

.1944 

5.1446 

.9816 

1.3788 

79 

12 

.2094 

.2079 

.2126 

4.7046 

.9781 

1.3614 

78 

13 

.2269 

.2250 

.2309 

4.3315 

.9744 

1.3439 

77 

14 

.2443 

.2419 

.2493 

4.0108 

.9703 

1.3265 

76 

15 

.2618 

.2588 

.2679 

3.7321 

.9659 

1.3090 

75 

16 

.2793 

.2756 

.2867 

3.4874 

.9613 

1.2915 

74 

IT 

.2967 

.2924 

.3057 

3.2709 

.9563 

1.2741 

73 

IS 

.3142 

.3090 

.3249 

3.0777 

.9511 

1.2566 

72 

19 

.3316 

.3256 

.3443 

2.9042 

.9455 

1.2392 

71 

20 

.3491 

.3420 

.3640 

2.7475 

.9397 

1.2217 

70 

21 

.3665 

.3584 

.3839 

2.6051 

.9336 

1.2043 

69 

22 

.3840 

.3746 

.4040 

2.4751 

.9272 

1.1868 

68 

23 

.4014 

.3907 

.4245 

2.3559 

.9205 

1.1694 

67 

24 

.4189 

.4067 

.4452 

2.2460 

.9135 

1.1519 

66 

25 

.4363 

.4226 

.4663 

2.1445 

.9063 

1.1345 

65 

26 

.4538 

.4384 

.4877 

2.0503 

.8988 

1.1170 

64 

27 

.4712 

.4540 

.5095 

1.9626 

.8910 

1.0996 

63 

28 

.4887 

.4695 

.5317 

1.8807 

.8880 

1.0821 

62 

29 

.5061 

.4848 

.5543 

1.8040 

.8746 

1.0647 

61 

30 

.5236 

.5000 

.5774 

1.7321 

.8660 

1.0472 

60 

31 

.5411 

.5150 

.6009 

1.6643 

.8572 

1.0297 

59 

32 

.5585 

.5299 

.6249 

1.6003 

.8480 

1.0123 

58 

33 

.5760 

.5446 

.6494 

1.5399 

.8387 

.9948 

57 

34 

.5934 

.5592 

.6745 

1.4826 

.8290 

.9774 

56 

85 

.6109 

.5736 

.7002 

1.4281 

.8192 

.9599 

55 

36 

.6283 

.5878 

.7265 

1.3764 

.8090 

.9425 

54 

37 

.6458 

.6018 

.7536 

1.3270 

.7986 

.9250 

58 

38 

.6632 

.6157 

.7813 

1.2799 

.7880 

.9076 

52 

39 

.6807 

.6293 

.8098 

1.2349 

.7771 

.8901 

51 

40 

.6981 

.6428 

.8391 

1.1918 

.7660 

.8727 

50 

41 

.7156 

.6561 

.8693 

1.1504 

.7547 

.8552 

49 

42 

.7330 

.6691 

.9004 

1.1106 

.7431 

.8378 

48 

43 

.7505 

.6820 

.9325 

1.0724 

.7314 

.8203 

47 

44 

.7679 

.6947 

.9657 

1.0355 

.7193 

.8029 

46 

45 

.7854 

.7071 

1.0000 

1.0000 

.7071 

.7854 

45» 

Cosine 

Cotangent 

Tangent 

Sine 

Eadians 

Angle 

262 


TABLES 


LOGARITHMS 


0 

1 

2 

3 

4 

5 

6 

7 

8 

4 

123 

456 

281 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

4  8  12 

17  21  25 

29  33  37 

11 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

4  8  U 

15  19  23 

26  30  34 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

3  7  10 

14  17  21 

24  28  31 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

3  6  10 

13  16  19 

23  26  29 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

369 

12  15  18 

21  24  27 

15 

1761 

1790 

1818 

1847 

1S75 

1903 

1931 

1959 

1987 

2014 

368 

11  14  17 

20  22  25 

16 

2041 

2068 

2095 

"2122 

2148 

2175 

2201 

2227 

1253 

2279 

358 

11  13  16 

18  21  24 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

257 

10  12  15 

17  20  22 

18 

•255:$ 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

257 

9  12  14 

16  19  21 

19 

278S 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

247 

9  11  13 

16  18  20 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

246 

8  11  13 

15  17  19 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

246 

8  10  12 

14  16  18 

22 

3424 

3444 

3464 

34S3 

3502 

3522 

3541 

3560 

3579 

3598 

246 

8  10  12 

14  15  17 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

37C6 

3784 

246 

7  9  11 

13  15  17 

24 

3802 

3820 

3838 

3856 

3S74 

3892 

3909 

3927 

3945 

3962 

245 

7  9  11 

12  14  16 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

235 

7  9  10 

12  14  15 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

235 

7  8  10 

11  13  15 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

235 

689 

11  13  14 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

235 

689 

11  12  14 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

1  3  4 

679 

10  12  13 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

1  3  4 

679 

10  11  13 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

1  3  4 

678 

10  11  12 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

1  3  4 

578 

9  11  12 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

1  3  4 

568 

9  10  12 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

1  3  4 

568 

9  10  11 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

124 

567 

9  10  11 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

1  2  4 

567 

8  10  11 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

57<5H 

5775 

5786 

1  2  3 

567 

8  9  10 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

123 

567 

8  9  10 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

1  2  8 

457 

8  9  10 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

1  2  3 

456 

8  9  10 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

1  2  3 

456 

789 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

1  2  3 

456 

789 

43 

IW35 

(J345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

1  2  3 

456 

789 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

1  2  3 

456 

789 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

1  2  3 

456 

789 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

123 

456 

778 

47 

6721 

6780 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

1  2  3 

455 

678 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

123 

445 

678 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

1  2  3 

445 

678 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

123 

345 

678 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

1  2  3 

345 

678 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

1  2  2 

345 

677 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

1  2  2 

345 

667 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

122 

345 

6  6  7 

TABLES 


263 


LOGARITHMS 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

123 

456 

789 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

122 

345 

567 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

1    2    2 

345 

567 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

1     2    2 

345 

567 

58 

7(184 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

1     1    2 

344 

567 

59 

77U9 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

1     1    2 

344 

567 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7882 

7839 

7846 

112 

344 

566 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

112 

344 

566 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

1     1    2 

334 

566 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

1     1    2 

384 

556 

64 

8062 

8069 

8075 

8082 

8(189 

8096 

8102 

8109 

8116 

8122 

112 

334 

556 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

1    1    2 

334 

556 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

1    1    2 

334 

556 

67 

8261 

3267 

8274 

8280 

8287 

8293 

8299 

8806 

8312 

8319 

112 

334 

556 

63 

S325 

3331 

SMS 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

1     1    2 

334 

456 

69 

8388 

S895 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

1    1    2 

284 

456 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

112 

234 

456 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

S555 

8561 

8567 

1    1    2 

234 

455 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

112 

284 

455 

73 

SIKM 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

1     1    2 

234 

455 

74 

S  692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

1     1    2 

234 

455 

75 

8751 

S756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

112 

233 

455 

76 

8808 

SS14 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

112 

233 

455 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

1     1    2 

238 

445 

78 

8921 

8927 

8932 

S98S 

8943 

8949 

8954 

8960 

8965 

8971 

1     1     2 

233 

445 

79 

8976 

S9S2 

8987 

s9!>8 

8998 

9004 

9009 

9015 

9020 

9025 

1     1    2 

233 

4.   4    5 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

1    1    2 

233 

445 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

112 

283 

445 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

1    1    2 

233 

445 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

112 

233 

445 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

112 

238 

445 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

1    1    2 

233 

445 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

1     1    2 

238 

445 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

Oil 

228 

344 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

Oil 

228 

344 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

0    1    1 

228 

344 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

Oil 

223 

344 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

0    1    1 

223 

344 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

Oil 

223 

344 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

Oil 

223 

844 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

Oil 

223 

344 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

0    1    1 

223 

344 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

Oil 

228 

344 

97 

9868 

9872 

9877 

9881 

98S6 

9890 

9894 

9899 

9903 

9908 

0    1    1 

223 

344 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

Oil 

228 

344 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

Oil 

223 

334 

INDEX 


Absolute  units,  23. 

Acceleration,  20 ;  angular,  31,33, 102  • 
central,  34,  82  ;  of  centre  of  mass, 
112  ;  dimensions  of,  23  ;  of  falling 
body,  22,  68,  259;  in  line  of 
motion,  21  ;  in  simple  harmonic 
motion,  40. 

Activity,  135. 

Amplitude  of  a  simple  harmonic 
motion,  41. 

Analytical  method,  13. 

Angle,  units  of,  4 ;  of  contact, 
228,  260. 

Angular  acceleration,  31,  33,  102. 

Angular  momentum,  102,  109,  159. 

Angular  simple  harmonic  motion, 
166. 

Angular  velocity,  31. 

Archimedes'  principle,  211. 

Atmosphere,  239,  241. 

Axes  of  coordinates,  7. 

Axis  of  couple,  127. 

Barometer,  239. 
Beam  compass,  2. 
Beam,  flexure  of,  199. 
Blackburn's  pendulum,  53. 
Boyle's  Law,   243;    deduced  from 

kinetic    theory    of   gases,    247 ; 

deviations  from,  245. 
Bulk  modulus,  190. 

Capillarity,  223,  230-235 ;  effect  on 

barometer,  241. 
Centre  of  gravity,  127. 


Centre  of  mass,  104  ;  acceleration 
and  velocity  of,  112  ;  kinetic  en- 
ergy of  motion  of,  146. 

Centre  of  oscillation,  172. 

Centre  of  percussion,  175. 

Centre  of  parallel  forces,  121. 

Centrifugal  force,  82,  111,  164. 

Circle  of  reference,  41. 

Clock  circuit,  86. 

Coefficient,  of  compressibility,  190  ; 
of  friction,  89 ;  of  restitution, 
155. 

Component,  of  acceleration,  21  ;  of 
displacement,  12,  of  force,  70 ; 
of  velocity,  18. 

Composition,  of  accelerations,  21 ;  of 
angular  velocities  and  acceler- 
ations, 33  ;  of  displacements,  8-15, 
of  forces,  70;  of  parallel  forces; 
120-126;  of  velocities,  18. 

Compound  pendulum,  171. 

Compressibility,  190. 

Conical  pendulum,  86. 

Coordinates,  7. 

Conservation,  of  angular  momen- 
tum, 109-112  ;  of  energy,  151 ;  of 
momentum,  78. 

Conservative  forces,  150. 

Couples,  126. 

Curvature,  radius  and  circle  of,  35. 

Curve  of  speed,  29. 

Curved  path  of  ball,  219. 

D'Alembert's  principle,  115. 
Day,  4. 


INDEX 


265 


Degree,  4. 

Density,  207,  259 ;  and  specific  grav- 
ity, 212. 

Derived  units,  23. 

Diffusion,  of  gases,  251 ;  of  liquids, 
236. 

Dimensions  of  space,  7. 

Dimensions  of  units,  of  acceleration 
and  velocity,  23 ;  of  force,  92  ;  of 
kinetic  energy,  138  ;  of  potential 
energy,  139 ;  of  work,  135. 

Displacements,  8-15  ,  in  simple 
harmonic  motion,  40. 

Dissipation  of  energy,  153 ;  of  ro- 
tation, 159 ;  on  impact,  156. 

Dyne,  62. 

Effusion  of  gases,  250. 

Elastic  hysteresis,  203. 

Elastic  lag,  203. 

Elastic  limits,  188. 

Elasticity,  187  ;  fatigue  of,  203  ; 
Hooke's  Law  of,  69, 168,  189  ;  im- 
perfections of,  202  ;  moduli  of, 
190,  201  ;  of  gas,  246. 

Energy,  133-164  ;  conservation  of, 
151  ;  dissipation  of,  153  ;  equiva- 
lence of  kinetic  and  potential,  139  ; 
kinetic,  137  ;  potential,  138 ; 
of  rotating  body,  114 ;  of  strain, 
136  ;  surface,  227. 

Epoch,  38,  4C 

Equilibrium,  of  a  body,  128,  129  ;  of 
floating  bodies,  215 ;  of  a  parti- 
cle, 73 ;  stable,  unstable,  neutral, 
143. 

Erg,  135. 

External  forces,  109,  113. 

Flexure,  199. 

Flow  of  a  liquid,  from  an  ori- 
fice, 215 ;  past  an  obstruction, 
218;  through  a  capillary  tube, 
223. 

Fluid,  205  ;  pressure,  205. 

Foot-pound,  135. 


Force,  59 ;  and  acceleration,  63 ; 
conservative  and  dissipative,  150  ; 
dimensions  of,  92  ;  effective,  115 ; 
external,  109,  113  ;  in  simple  har- 
monic motion,  68  ;  intermolecular, 
223  ;  internal,  109,  113,  140 ;  units 
of,  61,  63. 

Frequency,  31,  56. 

Friction,  88,  115. 

Gases,  238. 
Gramme,  62. 
Gravitation,  67,  86. 
Gyroscope,  177. 

Harmonic    motion,     39;     angular, 

166. 

Heat,  152. 

Hooke's  Law,  69,  168,  189. 
Horse-power,  136. 
Hydraulic  press,  210. 
Hydrometers,  212. 

Impact,  154. 
Impulse,  60. 
Inclined  plane.  89. 
Inertia,  60. 
Isotropic  bodies,  154. 

Joule,  135. 

Kilogramme,  61. 

Kinetic  theory  of  gases,  247. 

Laws  of  motion,  59,  65,  76. 
Liquid,  204. 

Mass,  62,  79 ;  centre  of,  104. 

Metacentre,  215. 

Metre,  1. 

Micrometer  caliper,  4. 

Modulus,  of  elasticity,  190,  195 ;  of 

gas,  246  ;  relation  between  moduli, 

201. 
Moment  of  force,  94,   102 ;    work 

done  by,  148. 


266 


INDEX 


Moments  of  inertia,  94,  97-101 ; 
comparison  of,  by  torsional  pen- 
dulum, 169. 

Momentum,  60,  78 ;  angular,  102, 
109,  159;  dimensions  of,  92. 

Neutral  surface,  199. 

Newton,    gravitation,   67 ;    impact, 

154 ;  laws  of  motion,  59,  65,  75 ; 

pendulum  experiments,  62. 

Origin,  8. 

Oscillation,  centre  of,  172 ;  of  gyro- 
scope, 180. 
Osmosis,  236. 

Parallel  forces,  120. 

Pascal's  principle,  209. 

Path  of  projectile,  26  ;  curved  path 
of  ball,  219. 

Pendulum,  Blackburn's,  53 ;  com- 
pound, 171 ;  conical,  86  ;  equiva- 
lent simple,  171;  reversible,  172; 
simple,  45  ;  torsional,  167. 

Periodic  motion,  38  ;  of  rigid  bodies, 
166. 

Phase,  38,  44. 

Poisson's  ratio,  196. 

Position,  7. 

Power,  135. 

Precession,  179. 

Pressure,  and  speed,  218  ;  of  atmos- 
phere, 239;  of  fluid,  205;  trans- 
missibility  of,  209 ;  on  curved 
surface,  229. 

Projectile,  26. 

Projection  of  a  simple  harmonic 
motion,  49. 

Pumps,  for  liquids,  242 ;  for  gases, 
249. 

Radian,  4. 

Radius,  of  curvature,  35 ;  of  gyra- 
tion, 101. 
Ranee  of  molecular  forces.  223. 


Rectangular  coordinates,  7. 

Resolution,  of  accelerations,  21  ;  of 
displacements,  12  ;  of  forces,  70 ; 
of  velocities,  18  ;  of  circular  mo- 
tion, 44. 

Rest,  18. 

Restitution,  coefficient  of,  155. 

Rigidity,  190. 

Rotation,  of  rigid  body,  96 ;  and 
translation,  114. 

Scalar  quantity,  15. 

Second,  mean  solar,  4. 

Shear,  185. 

Shearing  stress,  191. 

Simple  harmonic  motion,  39,  68 ; 
angular,  166. 

Siphon,  243. 

Solid,  185. 

Specific  gravity,  212. 

Speed,  18. 

Spring,  calibration  of,  69. 

Squeeze,  186. 

Strain,  185  ;  homogeneous,  186  ;  po- 
tential energy  of,  201. 

Stress,  77,  188. 

Surface  tension,  225 ;  measurement 
of,  236. 

Surface  Energy,  227. 

Thrust,  196,  210. 

Torricelli's  theorem,  216. 

Torsion,  191-195 ;  pendulum,  167. 

Translation,  10 ;  and  rotation, 
114. 

Triangle,  of  accelerations,  21 ;  of  dis- 
placements, 10  ;  of  forces,  73  ;  of 
velocities,  19. 

Tuning-fork,  46, 189. 

Uniform  circular  motion,  30;  force 
in,  82. 

Units,  of  angle,  4 ;  of  energy,  138  ;  of 
force,  61  ;  fundamental  and  de- 
rived, 23 ;  of  length,  1  ;  of  mass, 
61 :  of  time.  4  :  of  work.  135. 


INDEX 


267 


Vector  quantities,  15. 

Velocity,  angular,  31 ;  instantaneous, 
19 ;  in  simple  harmonic  motion, 
40 ;  of  centre  of  mass,  112 ;  uni- 
form, 18  ;  variable,  19. 

Vena  contracta,  216. 

Vernier,  3. 

Viscosity,  219. 


Watt,  135. 

Weight,  62. 

Work,  132;  diagram  of,  136;  di- 
mensions and  units  of,  125  ;  done 
by  moment  of  force,  148  j  rate  of 
doing,  135. 

Yard,  2. 

Young's  modulus,  195. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


FEB  191948 


l8Apr'§lLi) 


LC 

MAR    719& 

21-100m-9,'47(A5702sl6)476 


Ri_C'D  LD 

MAY  9     1961 


C'D  LD 

JANS    1963 


0981 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


